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ELF > X� @ @ UH��}��E�� �E�;E�}�E�H�H�� �� ]� u<v or u=v iff u<=v show equation again Coefficient of x^2 must be zero. Too difficult. You can still try numerical or graphical solution. That's the simplest form MathXpert can find. ✅ That's the answer. Sorry, that operation can't be applied here. Version Go to the MathXpert starting page You can use the green slider; click a white slider (slowly) to turn it green. The slider changes the graph immediately That polynomial is irreducible. That's the best MathXpert can do. How many terms do you want to see? Enter value of Prove by induction on which variable? Enter starting value of Definite Integration Enter your substitution (use = ) Write it as a function of what? Enter two expressions separated by a comma: Enter your proposed factor: Linear in which variable? Enter one of groups: Write it as a polynomial in what expression? Add what to both sides? Subtract what from both sides? Transfer what term left to right? Transfer what term right to left? Multiply both sides by what? Divide both sides by what? Work on which equation? Raise both sides to what power? Enter $a$ in $u=v => a^u = a^v$: Take what root of both sides? Apply what function to both sides? Evaluate at what value of Enter values of Find root between what two numbers? Add eqn no. ? to eqn no. ? Subtract eqn no. ? from eqn no. ? Multiply equation number ? by ? Divide equation number ? by ? Add ? times eqn no. ? to eqn no. ? Subtract ? times eqn no. ? from eqn ? Substitute for which variable? Swap eqns ? and ? (enter 2 numbers) Regard which variable as constant? Swap rows ? and ? Add row ? to row ? Subtract row ? from row ? Multiply row ? by ? Divide row ? by ? Subtract ? times eqn no. ? from eqn no. ? Raise both sides to what positive odd power? Raise both sides to what non-negative odd power? Enter a in a^u < a^v : Enter new base of logarithms: Enter u in lim uv = lim u lim v : Multiply num and denom by what? Divide num and denom by what? Enter u like this: u = x/h Enter the term u to move to the denominator Write as polynomial in what expression? Cannot determine the sign of the constant term. Enter your choice of u, the part to differentiate: Enter the intermediate point (or points) b: Enter your substitution like this: u = sin x Enter your definition like this: f(x) = 1/x Enter new definition like this: f(x) = 1/x Enter name of function to undefine. Example: f Enter initial values of Enter z-interval between contours: Enter min and max values of You could enter for example = n \pi; or just n \pi Enter new parameter value: Enter new value of the active parameter: Enter new parameter increment: Enter new increment for the active parameter: You could enter for example 1/ Enter maximum value of Please enter an equation number between 1 and Entries should be separated by a comma. Enter exactly TWO terms, please. Enter a variable name, please. That variable doesn't occur in this line. That expression evaluates to zero; try again. That expression is equal to zero. Please try again. Enter an integer, please. Left side of = must be a variable already in use. You don't have that many equations. Try again, please. Please enter two DIFFERENT equation numbers. Please enter a NUMBER. Enter exactly TWO numbers, please. One of your entries isn't a number. Try again, please. Please enter a POSITIVE number. Please enter a NONZERO number. Please enter TWO items, separated by a comma. Your first entry isn't a valid equation number. Your second entry is zero. Please try again. POSITIVE expression expected. NEGATIVE expression expected. NONNEGATIVE expression expected. NONPOSITIVE expression expected. Entries must evaluate to numbers. Enter two DIFFERENT numbers, please. First number must be < second number. Please use = sign. Enter VARIABLE = EXPRESSION. Left side of = must be a variable name. Please separate your three items by commas. Please enter THREE items, separated by commas. Enter a positive integer, please. Can't understand your entry Not a function you can use here. Try again, please. The largest acceptable integer is 32,768. Variable already in use. Choose a NEW variable. Right-hand side may not contain new variables. Positive ODD integer expected. You must enter a product Complex numbers are turned off, so 'i' is not allowed. Wrong number of initial values Please enter exactly 4 numbers. Right-hand side may not contain * Left-hand side must be the old variable of integration. Please use = sign. Example: f(x) = 1/x. Only variables are allowed as arguments on the left. Functions can have at most 12 arguments. Enter an inequality involving only the independent variable. Use < and <= only to define your interval. Your input does not define an interval. Left side of = must be a variable. No new variables can be introduced here. Left side of = must be a derivative. You already have a function by that name. Too many defined functions, can't accept another one. That function isn't defined, so you can't redefine it. That function is pre-defined, you can't define it. Repeated variable on the left side. Extra variable on the right side. Your second entry is not a valid equation number. Second entry must be a variable name. No more than two variables allowed in simultaneous substitutions. Illegal symbols on the right. Please enter a constant. Because $a^n=b^n$ doesn't imply $a=b$, this equation may not be equivalent to the previous one. Check your final solutions in the original equation. ln $z$ defined only for positive $z$ $$log(b,z)$$ defined only for positive $z$ That would result in an undefined term Can't take $^n\sqrt a$ unless n is a positive integer. Can't take even root of negative number. take $\sqrt $ of both sides take $^n\sqrt $ of both sides de Moivre's theorem Left side would not be defined. change signs Right side would not be defined. apply !Solutions all check OK in original equation for !None of the solutions satisfy the original equation for Reject all solutions. !Only one solution satisfies the original equation for Check solutions !Keep only solutions that satisfy the original equation. Original problem not an equation or inequality. Equation not solved yet, can't check solutions. !Solution checks in the original equation for Check solution !Solution doesn't satisfy the original equation for 1/u can't be zero Value not in range of complex $\sqrt $ function Value not in range of complex root function Square roots must be non-negative Even roots must be non-negative Roots can't be negative. Roots must have non-negative imaginary part. add to both sides add subtract 0 subtract subtract term multiply by 1 multiply both sides multiply by divide by 1 Can't divide by zero divide both sides divide by if $a=b$ then $b=a$ if $a<b$ then $b>a$ if $a>b$ then $b<a$ if $a\le b$ then $b\ge a$ if $a\ge b$ then $b\le a$ add 0 a^x not zero if $a>0$ e^x is never 0 e^x > 0 a^x > 0 if $a>0$ !One factor has to be nonzero by current assumptions !Some factors are nonzero by current assumptions !Equation is an identity zero factor $a=0$ or $b=0$ if $ab=0$ selected You already have an equation selected. First use 'show all equations'; after that you can select an equation. show all equations combine multiplicities combine solutions You must first evaluate the derivative. You must first evaluate the integral. You probably want 'experiment numerically' on the LIMITS menu. This operation doesn't work on limits. Definite integrals can be evaluated numerically, but not by this operation. Look on the DEFINITE INTEGRALS menu. evaluate at point solve numerically root not bracketed Apparent failure of numerical method to converge, even after many iterations. Left-hand side of equation must be quadratic. Right-hand side of equation must be constant. complete the square subtract right side cancel common factor common factor is zero |u|=c iff u=c or u=-c cross multiply unsolvable equation cancel constant solve linear equation if a=b then $a^n=b^n$ if a=b then c^a=c^b u=v iff u^2=v^2, $uv\ge 0$ drop nonzero factor drop nonzero factors You have complex numbers turned off. discriminant is negative quadratic formula factor integer That integer is prime Integer (or integers) are prime No factors less than than 65,536, but definitely not a prime. and probably is prime. Largest factor has no factors less than 65,536, but it definitely isn't prime No factors less than 65,536 That operation factors only integers. complex prime factors that number has no complex factors complex integer has no complex factors Large number has no factors with real and imag parts less than 65,536. No factors with real and imaginary parts less than 65,536 That operation requires nonzero real part. That operation requires nonzero imaginary part. factor n+mi factor coefficients factor denominator Denominator too hard to factor Denominator cannot be factored factor expression denom You must first evaluate the limit in the denominator. integral in the denominator. Use $a(b/c) = ab/c$ to multiply by a rational fraction. zero denominator invert and multiply common denominator $|u|=u$ if $u\ge 0$ $|a|= -a$ if $a\le 0$ $|u|=-u$ if $u\le 0$ definition of $|u|$ Maybe you meant $\le $ instead of < $|u|<v$ iff $-v < u < v$ Maybe you meant < instead of $\le $ $|u|\le v$ iff $-v \le u \le v$ $u<|v|$ iff $v<-u$ or $u<v$ $u\le |v|$ iff $v\le -u$ or $u\le v$ $|u|^(2n)=u^(2n)$ requires $u$ to be real $|u|^(2n) = u^(2n)$ $|u^n|=|u|^n$ (u real) $u=v$ iff $u^2=v^2$, $uv\ge 0$ $|a||b|=|ab|$ $|uv|=|u||v|$ $|cu| = c|u|$ if $c \ge 0$ $|u| = c$ if $c\ge 0$ $0 \le |u|$ is true $|u| \ge 0$ is true $|u| < 0$ is false $0 > |u|$ is false $|u|/u=1$ iff $0 < u$ $u/|u|=1$ iff $0 < u$ $|u|/u=-1$ iff u<0 $u/|u|=-1$ iff u<0 $|u|/u=c$ => $c=\pm 1$ You must first make one side zero. Complex numbers are turned on, and there will be complex roots even when b^2-4ac < 0. Expression is not quadratic. discriminant < 0 Negative discriminant implies no solution Too many factors That doesn't multiply out correctly. factor quadratic This quadratic can't be factored using integer coefficients. Number of trials: Trying linear factor: Coefficients or degrees too large. Sorry about that. Numerical coefficients must be integers. Numerical coefficients must be complex integers. Trying factors: Expression not a polynomial. Coefficients must be integers This operation only works if there are at most two variables. In $^n\sqrt (a^n) = a$, $a$ must be non-negative. First factor out a common term. Constant term is too big for MathXpert to factor. Leading coefficient is too big Exhaustive search failed. There is no linear factor. But the constant term was not a monomial. Maybe it factors; if so, we missed some possibilities. linear factor polynomial division That would create a very long expression. Try a problem with an exponent less than 100. Proposed factor must contain a variable. Requires a sum of 4 or more terms Sum of terms expected. Too many terms in the group. The group you entered contains a term which isn't in the problem. Perhaps you made a typing mistake. The group you entered is not a polynomial. factor by grouping expand sin Numerical factoring works only on polynomials in only one variable. Coefficient too large to handle calculate roots polynomials, i.e. sums of monomials. Can't compute roots numerically Impossible equation Solve equation variables to left constants to right transfer terms That operation works only on systems of linear equations. Variables already lined up. First move constants to the right side Before lining up the variables, collect terms. line up variables Add eqn to eqn Sub eqn from eqn Multiply eqn by constant Divide eqn Add times eqn Subtract Substitute for Exponent must be constant for this to work. swap equations That operation is only for use after the equations are already all solved. rearrange equations rearrange drop identity drop identities is already treated is constant. Equation is not solvable. That would make all the equations constant Regard as constant !From now on is constant. contradiction No solution You must first evaluate the limit(s) Can't verify that the integral is defined and finite Can't apply $x\cdot 0 = 0$ when x is not defined. regroup factors bring number to front arithmetic numbers multiply to 1 collect numbers collect powers Use multiply out if you want to multiply out a product of more than two sums. To expand a power, use the binomial theorem. multiply out Sorry, the exponent has to be less than two billion. binomial theorem multiply and cancel put factors in order put terms in order You must factor the denominator first. partial fractions common factor regroup terms cancel $\pm $ terms cancel collect and cancel collect like terms cancel terms undefined expression $0^0$ is undefined The law $(a^b)^c = a^(bc)$ is valid only when $a>0$ or $c$ is an integer. For example, $((-1)^2)^(\onehalf)$ is 1, not -1. MathXpert can't cope with $root(n,x)$ unless $n < 65537.$ undefined power Exponent must be positive. Exponent not known to be defined This power of -1 is undefined when working with real numbers only, as you are at the moment. Try it again under Complex Numbers (-1) to a power (odd/odd) = -1 $$root(n,b^n)=b$$ ($n$ odd) $$root(n,b^n)=b$$ if $b\ge 0$ or n odd $$root(n,c^nb)=c root(n,b)$$ ($n$ odd) $$root(n,a^nb)= a root(n,b)$$ ($a\ge 0$ or n odd) compute complex root compute square root compute root requires $x\ge 0$ for even $n$ requires $a\ge 0$ and $b\ge 0$ undefined root !Equation contradicts assumptions This operation only works if there's an exponent on one side of the inequality. One side of the inequality must be an odd power for this to work. You can't take roots of inequalities with complex numbers turned on. This only works if the exponent is odd, or both sides of the inequality are >= 0. Left side is negative, so that won't work. That only works on polynomials in one variable. Coefficients must be integers. Can't identify any positive factor. divide num and denom $^n\sqrt u < v => u < v^n$ $^n\sqrt u \le v => u \le v^n$ $u < v => u^n < v^n$ (n odd, n>0) $u \le v => u^n \le v^n$ (n odd, $n\ge 0$) solve linear ineq examine the signs of the factors Your equations are nonlinear. You must first collect like terms. matrix form Multiplying by 1 won't change anything. Add row to row Sub row from row Multiply row Can't use Cramer's rule unless the number of equations is the same as the number of variables. Cramer's rule evaluate determinant square-free factors Work on which inequality? Work on which equation or inequality? drop positive factors Can't verify denominator is positive Can't verify numerator is positive Can't verify sides of inequality are non-negative $0\le u<v => u^n<v^n (n>0)$ $0\le u\le v => u^n\le v^n (n>0)$ Raise both sides to what positive power? Can't evaluate numerically because a variable is present. Answer would be indeterminate. Maybe common denominators would help. The entire expression inside 'lim' is independent of Exponent isn't constant so that operation can't be used. Exponent is zero. Simplify it directly. Resulting limit would be undefined Index of root is not odd $lim ^n\sqrt u = ^n\sqrt (lim u)$ if $n$ is odd Limit under $^n\sqrt $ would not be positive, $lim ^n\sqrt u = ^n\sqrt (lim u)$ if lim u > 0 Limit under $\sqrt $ would not be positive, $lim \sqrt u = \sqrt (lim u)$ if lim u > 0 limit of polynomial $$lim(x->a,f(x))= f(a)$$ (polynomial f) Can't find the term you entered as a common factor. Use factoring instead to take that term out. Only one factor is divisible by that term. Enter the common factor (a in the formula): You can't factor out 1. First you must evaluate the derivative. definition integral derivative sum limit indexed product defn of expand defn Too many expressions separated by commas. At most two at a time can be handled. write as function of It wouldn't help to substitute for a constant. That substitution would not eliminate a variable so afterwards you would have MORE variables. MathXpert can't handle any more variables! MathXpert can't handle so many variables! Can't use a bound variable for induction. !Prove it by induction on It's already proved. Basis case is already done. Trying a proof by nested induction. You can undo if you didn't try nested induction on purpose. basis case Induction step is already done. induction step This is to be proved using the induction hypothesis. proved by induction Some assumption depends on the induction variable. Possibly what you are trying to prove isn't correct without more assumptions. You aren't working on an induction step now, so you don't have an induction hypothesis. induction hypothesis No variable present, can't use induction. More than one possible induction variable. First use 'select induction variable' Numerator isn't constant calculate limit of rational function Limit of denominator is zero undefined Limit in numerator would be undefined. Limit in denominator would be undefined. This limit is not an indeterminate form. MathXpert cannot determine whether this is an indeterminate form or not. Sorry, but under those circumstances the rule can't be applied. MathXpert cannot evaluate the limit that would result. L'Hospital's rule That only works on limits of quotients. mult num and denom In $du/dx$, $x$ must be a variable. defn of derivative That operation won't work here because the expression inside the function isn't exactly the same as the independent variable. Try the chain rule version of the operation instead. The exponent depends on so you can't use that operation. power rule dc/dx=0 (c constant) product rule $$diff(c/x,x) = -c/x^2$$ (c constant) $$diff(c/v,x) = -c/v^2 diff(v,x)$$ (c constant) quotient rule in $d/dx$, $x$ must be a variable. Can't take derivatives of non-integer order fundamental theorem of calculus Can't differentiate while solving equations. Example: solve $x = 1$; $dx/dx = 1 = d1/dx = 0$ definition of $d^nu/dx^n$ calculate derivative ✅ That's a good answer. To see more derivatives, choose Differentiate the Equation. $\sqrt x\sqrt y = \sqrt (xy)$ requires $x\ge 0$ and $y\ge 0$ Not linear. You cannot solve nonlinear systems with MathXpert. Expression to be evaluated: Integral to be evaluated: System of differential equations to solve: Functions to be graphed : Does depend on Which variables depend on ? (enter at least You must enter a variable. Press any key and try again. isn't in the problem. Enter a variable now. Press Enter again to confirm. You must enter one or more variables. Press any key and try again. After specifying the constants, equations must be linear. Complex numbers allowed Using real numbers only $\sqrt $ and $^n\sqrt $ allowed in factors $\sqrt $ and $^n\sqrt $ not allowed in factors auto mode menu mode Solve for which variable? Differentiate with respect to which variable? Integrate with respect to which variable? Which is the independent complex variable? Which variable do you want on the horizontal axis? Which is the independent variable? Which variable do you want on the vertical axis? Can't evaluate left side Can't evaluate right side evaluate at point Base is not constant. Use $d/dx u^v = (d/dx) e^v^(ln)^u$ instead. The exponent is constant, so use the power rule instead. Can't take ln of a negative number. The two derivatives must be with respect to the same variable. differentiate polynomial $a$ must be positive in $ln a$ That operation requires the exponent to be real. That would create a log of a negative number. That would create ln of a negative number. The new equation may have solutions that are not solutions of the previous equation, since $log(ab)$ is defined when both $a$ and $b$ are negative, but $log a$ and $log b$ are not. Remember to check your final solution(s) in the original equation. compute complex log compute log compute complex ln compute ln Writing $e^n = e^n^(ln)^e$ can't possibly help. Introducing $ln u$ requires $u>0$, as complex numbers are turned off now. definition of exp change base of logs Sorry, the base is too large for MathXpert. factor out base Base is not an exact power express as polynomial differentiate the equation break $\int |f(t)| dt$ evaluate integral Exponent must be negative exponent of sec must be an integer exponent must be at least 3 reduce $\int sec^n t dt$ integrate matrix term by term combine constants Can't take ln of non-positive number Can't take ln of a negative number Your substitution did not mention a new variable. Only one new variable can be introduced. That substitution won't eliminate Can't solve substitution for Substitution either not one-to-one or just too complicated to handle. That substitution isn't well-defined. In solving for , an undefined term occurs. No substitution has been defined You must first compute $d$ Angle is not expressed in degrees. Can't do arithmetic with that angle multiple of odd multiple of Use 'angle is multiple of $90\deg $' instead. radians to degrees degrees to radians First express the angle in degrees. This only works on positive angles. angle = $30\deg + b 45\deg $ angle = $60\deg + b 45\deg $ $\int c dt=ct$ (c constant) Exponent depends on When the exponent is -1 the power rule does not apply. When the exponent is 1 that rule does not apply Can't verify that the interval of integration does not include a singularity. You can't integrate 1/ t across t=0. (t-a) across t=a. You must specify the interval before tabulating values. You can't change the interval, except by using undo back to where you rejected a point outside the interval. critical points add critical points endpoints add endpoints decimal to fraction Decimal numbers over 100,000 are assumed to be approximate, and cannot be converted to fractions. Decimal numbers less than 0.000001 are assumed to be approximate, and cannot be converted to fractions. This decimal number is not equal to any simple fraction. MathXpert treats most decimal numbers as inexact, approximate numbers. The manual, or Help, explains exactly which decimals can be converted to fractions. You can't apply that law when the exponent is not defined. You must first evaluate the limit in the exponent. You must first evaluate the integral in the exponent. (-1)^n = -1 (odd n) (-1)^n = 1 (even n) That operation is a good choice, but you should apply it to a different term than the one you selected. First evaluate the limit in the exponent to be sure that it is defined and positive. First evaluate the limit to be sure it is not zero. That would create an undefined power. The new numerator would be undefined. The new denominator would be undefined. mult num and denom by cos x - sin x mult num and denom by cos x + sin x mult num, denom by ? double angle formulas mult num and denom by $(^n\root c)^n=c$ if $n$ is odd $(^n\root c)^n=c$ (if defined) common denom and simp in Too many equations. Graph original equation or current equation? Graph original equation or current equations? Limit expected. MathXpert can't graph systems of linear equations in more than two variables. Graphing not available for inequalities Graphing not available for complex numbers Add selected equation to which equation? Cannot graph this expression. Subtract selected equation from which equation? Swap selected equation with which equation? Add selected row to which row? Subtract selected row from which row? previous expression Well, this substitution doesn't seem to work. Couldn't make that substitution work, so let's try something else. integrand=$f(u)\times du/dx$ Expression too big to display. You can click OK or Edit even though your formula is not visible. Trying substitution: substitution That substitution doesn't eliminate That integral can't be done using only substitution and table lookup. Can't compute lower limit of integral. The new point must be between the limits of integration. You must enter a term that divides the integrand MathXpert can't integrate $dv$ for that choice of $u$ so that choice of $u$ won't work. summary of work That operation almost applies here, but you must first prepare for it by using $ You are getting ahead of yourself here. Before you are ready to use that operation, Can't compute upper limit of integral. you have to prepare by using MathXpert gave up, sorry That operation won't work here, but you seem to be on the right track You might try $ integrate polynomial trial factor too big to display Number of terms evaluated: Computation stopped at user's request. Then $v = $ You should first differentiate the equation. After that, you can put in the numbers. substitute values numerical calculation Can't integrate numerically. There must be something unusual about this integral! $u(x) = -u(-x)$ isn't true for this integrand. Can't verify $u(x) = -u(-x)$ for this integrand. $u(x) = u(-x)$ isn't true for this integrand. Can't verify $u(x) = u(-x)$ for this integrand. It would be better to use integration by parts. Polynomial division tried on non-polynomials. Difference of upper and lower limit must be an integer Lower limit must be $\le $ upper limit Too many terms definition of $\sum $ Can't evaluate lower limit of integral. Can't evaluate upper limit of integral. specified value You must first rename one index variable compute binomial coef Lower limit must be less than upper limit split off last term You can't show more than 1000 terms expand cos The sum doesn't have that many terms. split off first terms More than 100 terms. Too long to be useful. 0! = 1 by defn 1! = 1 by defn defn of factorial rename index variable Limit of numerator is zero. Limit of denominator is not zero. Denominator must be of one sign near the limit. lim u/v undefined if $v\to 0$ and not $u\to 0$ Exponent in denominator must be positive Exponent must be positive $lim(1/u^n)$ is undefined if $u\to $0, $n$ odd Can't check limit is at an odd multiple of $\pi /2$. The one-sided limits have different signs. Can't check limit is at a multiple of $\pi $ Can't check limit is at an odd multiple of $\pi /2$ undefined limit First evaluate the other limits in the product. Can't verify other factors are nonzero. You have reached an indeterminate form. Undo back to where these terms were created. First evaluate the other limits in the sum. Can't verify that the other summands are defined. denominator must be defined and nonzero. denominator must be nonzero. !Sign of denominator unknown lim(u/v) = lim u if $lim v\neq 0, lim u =\pm \infty $ numerator must be defined and nonzero. numerator must be nonzero lim(u/v) = 0 if $lim u\neq 0, lim v =\pm \infty $ First evaluate the numerator MathXpert can't determine the sign of the numerator. Base must exceed 1 Base must be positive Base must be less than 1 and the limit of the numerator is not finite, Can't prove denominator is positive. Can't prove denominator is negative. Can't prove numerator is $\ge 0$. Can't prove numerator is $\le 0$. Index of root must be even. Index of root must be odd, or denominator positive. Index of root must be odd, or numerator $\ge 0$. Index of root must be an integer. The result would have the form zero times undefined. The result would be indeterminate. $\sqrt (ab) = \sqrt a\sqrt b$ if $a\ge 0$ $\sqrt (a^2b)=a\sqrt b$ if $a\ge 0$ undefined square root $(\sqrt a)^(2n) = a^n$ if $a\ge 0$ $\sqrt (a^(2n+1)) = a^n\sqrt a$ $\sqrt (a^2)= |a|$ $\sqrt (a^(2n))= |a|^n$ $\sqrt (a^(2n))= a^n$ if $a\ge 0$ $\sqrt (a/b) = \sqrt a/\sqrt b$ $\sqrt a/\sqrt b = \sqrt (a/b)$ $\sqrt x = x^(\onehalf)$ More than 50 terms in rationalizing factor-- factor integer known root multiply out and simplify $$lim(t->0,cos(1/t))$$ is undefined $$lim(t->0,sin(1/t))$$ is undefined $$lim(t->0,tan(1/t))$$ is undefined Can't verify hypotheses of squeeze theorem. squeeze theorem Suggested limit is zero, so that doesn't work. Suggested limit is infinite Can't verify suggested limit is defined and not zero. definition of e The substitution must be a monotonic function. change limit variable is continuous function not defined in neighorhood on the right on the left Can't prove that the numerator is zero Can't prove that the numerator is infinite $(sin t)/t \to $ 1 as $t\to $ $(tan t)/t \to $ $(1-cos t)/t\to $ 0 as $t\to $ $(cos(t)-1)/t\to $ $(1-cos t)/t^2\to \onehalf $ as $t\to $ | That wouldn't help: you already have $e$ to a power. What you entered does not divide the expression in the limit. $(ln(1+t))/t\to $ $(ln(1-t))/t\to $ -1 as $t\to $ $(e^t-1)/t\to $ Can't graph limits approaching an unspecified number. Equation is already solved. Maybe you want 'graph original equation'. Equations already solved. a/0 = undefined $a/0+ = \infty $ if a>0 $a/0- = -\infty $ if a>0 $a/0+ = -\infty $ if a<0 $a/0- = \infty $ if a<0 $\infty /0$ = undefined Both numerator and denominator approach 0, $1/u\to \infty $ if u>0 and $u\to 0$ $1/u\to -\infty $ if u<0 and $u\to 0$ lim 1/u undefined if lim u=0 Mathert can't calculate the limit of the denominator. $u/v\to \infty $ if $v lim u > 0$ and $v\to 0$ $u/v\to -\infty $ if $v lim u < 0$ and $u\to 0$ MathXpert can't calculate the limit of the numerator. (n odd and positive) MathXpert can't determine the sign of the exponent. n not odd in $^n\sqrt $ $^n\sqrt x\to -\infty $ as $x\to -\infty $, n odd $^n\sqrt x\to \infty $ as $x\to \infty $ $\sqrt x\to \infty $ as $x\to \infty $ $u^\infty = 0$ if 0 < u < 1 $1/x^n\to 0$ as $x\to \infty $ (n>0) $1/x^n\to 0$ as $x\to -\infty $ (n>0) $x^n\to \infty $ as $x\to \infty $ (n>0) $x^(2n)\to \infty $ as $x\to -\infty $ (n>0) $x^n\to -\infty $ as $x\to -\infty $ $1/u^(2n)\to \infty $ if $u\to 0$ $e^x\to \infty $ as $x\to \infty $ $e^x\to 0$ as $x\to -\infty $ undefined factor lim uv = $-\infty $ if $u\to -\infty $ and lim v \neq 0 lim uv = $\infty $ if $u\to \infty $ and lim v \neq 0 undefined summand $lim u+v = -\infty $ if $u\to -\infty $ and lim v \neq 0 $lim u+v = \infty $ if $u\to \infty $ and lim v finite $u^(-\infty) = 0$ if u > 1 $u^(-\infty) = \infty $ if 0<u<1 Exponent is too large Exponent must be an integer Instead use $$lim(t->a,c) = c$$. false true Enter an equation Equation too complicated to solve solve and substitute Expression too tall to display. Try another problem. Try this one: Is this what you meant? so we don't need to consider points where the derivative is undefined. !This function is everywhere differentiable, !This function is differentiable on the whole interval, !Can't simplify the expression for the points where f' is undefined. points where is undefined You must tabulate values first. limits at open ends No solution in the specified interval. drop values outside interval You should add the endpoints before you tabulate the values. Watch out for points where the derivative is undefined! You shouldn't tabulate values yet. You must first solve the equation(s) for tabulate values Can't tabulate at non-numerical values of select maximum select maxima select minimum select minima !No maximum on this interval !No minimum on this interval !No maximum !No minimum First consider the limits at the endpoint(s) where you now have 'undefined' You must first evaluate the limit. You've already chosen that once. You can't choose it again. You must specify an interval first. First solve the equation. Solve for Too many solutions in the interval. Next line would be very long. Eliminate integer parameter Answer cannot be guaranteed correct This operation can't be used in minmax problems. Try 'reject point outside interval' on minmax menu. !Reject solutions which contradict assumptions All points in the interval where $f'(x)=0$ are already listed. You must make a table of $y$-values first. If the min or max occurs at an endpoint, it will be a mistake to reject endpoints too soon. csc $u$ is never zero sec $u$ is never zero Too many such points in the interval. Sorry, can't calculate those points. arcsin $u$ defined only when $|u|\le 1$ 30-60-90 triangle 45-45-90 triangle sin u=1 iff $u=\pi /2+2n\pi $ sin u = 0 iff $u = n\pi $ sin u = -1 iff $u = 3\pi /2+2n\pi $ cos u=0 iff $u=(2n+1)\pi /2$ cos u = 1 iff $u=2n\pi $ cos u = -1 iff $u = (2n+1)\pi $ sin u = c iff $u=(-1)^narcsin c + n\pi $ arccos u defined only when $|u|\le 1$ cos u = c iff $u = \pm arccos c + 2n\pi $ tan u = c iff $u = arctan c + n\pi $ Resulting sum of limits would be indeterminate. Resulting limits would be too hard for MathXpert. MathXpert can't evaluate the sum. Evaluate it numerically. MathXpert cannot draw that graph, sorry. Please try something else. use the definition of definition of the independent variable? (If not it will be a parameter.) Function not defined anywhere on this interval. Identity not verified. Test it numerically. Identity is not valid. This operation won't introduce a root of a variable lim sec x undefined at $(2n+1)\pi /2$ Limit has different signs at $\pm \pi /2$ interval to consider No interval specified, using entire real line. $lim((u+a)/(v+b)) = lim(u/v)$ if $a/u\to 0$ and $b/v\to 0$. $lim((u+a)/v) = lim(u/v)$ if $a/u\to 0$. $lim(u/(v+b)) = lim(u/v)$ if $b/v\to 0$. lim(u+a) = lim(u) if $a/u\to 0$ $x^n\to \infty $ as $x\to \infty $ if n>0 First evaluate the limit in the denominator. Can't do arithmetic with zero in the denominator. Use the operations on the INFINITIES menu. Enter a numerical value for You must enter a numerical value. Numerical value too large or small. $(ab)^n=a^nb^n$ requires $a^n$ and $b^n$ defined. $cot \pi $ is undefined. Use the formula for tan(u-v) instead. That would result in an undefined value of tan. That would result in an undefined value of cot. Trig arguments already are fractions. $\sqrt (a^2b)=|a|\sqrt b$ $\sqrt (a^2)=|a|$ $|a|/|b| = |a/b|$ $|a/b| = |a|/|b|$ $|a/b| = |a|/b if b>0$ $tan u = 0$ iff $u = n\pi $ write in form $x+yi$ Expression too complicated, simplify it first Express complex log in polar form first. $ln ab = ln |a| + ln |b|$ $ln ab = ln a + ln b$ $log ab=log |a|+log |b|$ $log ab = log a+log b$ use leading terms Here $u$ is the leading term of the numerator. Here $v$ is the leading term of the denominator. Here $u$ and $v$ are the leading terms of num and denom. !Solutions found so far check in original equation for of the previous equation, since $log(a/b)$ is defined when both Introducing $log u$ requires $u>0$, log of both sides take ln of both sides Can't take log of a non-positive number $\int |t| dt = t|t|/2$ Rotate 30 degrees left Rotate 30 degrees right Rotate 30 degrees up OK Cancel Yes No Stop Computation Rotate down Increase initial value Decrease initial value Initial value of independent variable Entry is not a number Interval incorrectly specified odd integrand even integrand Input Error Searching for factors up to: You were working on a saved graph, so 'next problem' does not make sense. Evaluate limit Matrix is singular (or very nearly so); numerical inversion failed. compute inverse matrix not invertible Too many factors, MathXpert can't handle it. Left side isn't positive Pseudoprime test, base: More than one parameter is present, but only one is allowed. the problem That's the last one. No next problem That's the first one. No previous problem Type in a problem, or use Previous, or select a new problem source. Type in a problem, or use Next, Assumptions Definitions No assumptions Text and background colors are too similar. You'd better change one or the other. Highlight and background colors are too similar. Highlight and text colors are the same. Warning This will disable highlighting. Reason and background colors are too similar. Functions Which function do you want to undefine? MathXpert has no suggestion. Maybe you're finished. Hint Graph Line Graph Colors Singularities Jumps Expecting a number. Expecting a positive number. Expecting a number less than the upper bound. Expecting a number larger than the lower bound. Show Grid Hide Grid Show Titles Hide Titles Polar Coordinates Rectangular Coordinates Show Graph Tools Hide Graph Tools Hide Direction Field Show Direction Field Nonzero numbers must be at least 10 to the -12 Number cannot exceed 10 to the 12 Show Point and Slope Hide Point and Slope Intermediate numbers were too large or small Your graph may be incorrect MathXpert could not compute the singularities. Function nowhere defined on the interval shown. No graph visible Error in lower limit Error in upper limit Please enter both limits, or neither. Integral to be approximated: $$diff(u,t) = diff(v,t)$$ if $u=v$ MathXpert could not compute the roots. MathXpert could not compute the solution. Author's Commentary double dollars missing in commentary on line dollar sign missing in commentary on line Warning to author of commentary at line number No commentary available. Left endpoint must evaluate to a number. Right endpoint must evaluate to a number. Left endpoint must be less than right endpoint. Select the two independent variables Problem is not correctly formed. Line too long in problem file: 255 characters maximum. numbers. Upper and lower limits must be or evaluate to numbers. Help not available on that menu item. Sorry Enter a, the new exponent in (1/a) ln u^a That would create an indeterminate limit. Select TWO independent variables. Press and hold the Shift key while making the selections. Not enough memory to make a 3d graph. Failure Cannot initialize 3d graphics. Not enough memory to open another document. Close this calculation and start the next problem? Close this calculation and start the previous problem? Close this calculation? Close this graph? You have the problem source set to Type It In, and there is no next problem. That was the last problem, so there is no next problem. That was the first problem, so there is no previous problem. The requested problem has been typed incorrectly. This button cannot be used. The requested problem is inappropriate for this topic. This button cannot be used. This operation only factors out repeated factors, which do not occur in a root in this expression. Operation Does not Apply Cannot verify the condition, n odd or u non-negative. Zero denominator would be created. 0 < u/v => 0<v if u>0 $0 \le u/v$ => $0\le v$ if $u\ge 0$ $0 \le u/v$ => $0\le u$ if v>0 0 < u/v => 0<u if v>0 Solve equation ? for ? Can only solve for a variable. is already solved for Add ? to both sides of equation ? Subtract ? from both sides of equation ? Try again, please. Complex numbers are turned off, and that equation has no real solution. cancel term from both sides Add ? to both upper and lower limits clear denom of i complex arithmetic No applicable operations Try again, please Selection and text colors are too similar. Selection and highlighted text colors are too similar. Selection and background colors are too similar. Selection of overlapping terms is not allowed. Exponent is not odd. Can't verify exponent is odd. No real roots Confirmation cancel under $^n\sqrt $ cancel under $\sqrt $ make a substitution Cramer's rule would create a zero denominator. This means your equations are not independent, so they don't have a unique solution. Cramer's rule only works when the equations have a unique solution, which these don't. MathXpert cannot evaluate a large determinant containing variables. Matrix is not square, so you cannot take its inverse. MathXpert cannot invert matrices of dimension 4 or more unless all entries are numerical. MathXpert cannot invert this matrix, because the numbers involved are quite large or quite small. Matrix cannot be inverted, because its determinant is zero. Taking the matrix inverse only works when the equations have a unique solution, which these don't drop zero column drop zero row drop duplicate row if a^u = a^v then u=v This cubic has no quadratic term. Polynomial is not cubic. First eliminate the quadratic term by $u = x+b/3$ You must first compute the discriminant. cubic formula You have already computed the discriminant. Can't determine which quadrant this number is in. decimal value of $\pi $ decimal value of e Angle is already in degrees. Logarithm to be computed must not contain variables. Root to be computed must not contain variables. compute discriminant rationalize numerator rationalize denom find common denom f(undefined)=undefined That operation only factors a polynomial numerator or denominator. The numerator or denominator must be a sum of monomials. factor out in limit MathXpert could not compute the jumps. MathXpert could compute neither the singularities nor the jumps. Transforming a two-sided limit to a limit at infinity is illegal. factor out constant $\sum $ 1 = number of terms telescoping sum polynomial form Only one variable is allowed. Consider using 'write as polynomial (in ?)', or 'simplify sums and products'. That expression is already in polynomial form. MathXpert is unable to check the solution. Whether a solution checks or not may depend on the value of the parameters. compute function decimal value of $x^n$ decimal calculation complex decimal calc. There is a similar operation that works on fractions. elim compound fracts Constant must be nonzero, or remaining limit defined. Extra variables will be treated as constants. Notice You entered this problem under Simple Integration, but it may need integration by substitution or parts. It would be better to use the law $(a+b)/c = a/c + b/c$ on the integrand and then break up the problem into two integrals. Zoom out. If you have a touch screen or trackpad, you can also use pinch-to-zoom. Zoom in. If you have a touch screen or trackpad, you can also use pinch-to-zoom. Display assumptions Display singularities Display jump discontinuities Show author's remarks Change graph paper The resulting limit would not be finite or infinite, but would be undefined in some more complicated way. Under those circumstances L'Hospital's rule cannot be correctly applied. You must first evaluate the derivative or derivatives. One of your denominators will evaluate to zero. You can't use common denominators on such a fraction. Limitand is not defined at the limit point, and is therefore not continuous at the limit point. Equation not defined at left endpoint. Equation not defined at right endpoint. Perhaps your function is not continuous on the interval. Function must be defined and continuous on the interval. Even if it is, if extremely large or small numbers arise in the calculations, numerical solution can fail. a=0 or b=c if ab=ac Number entered is too large. Expansion would be too long. numerical integration You must first evaluate the derivative in the integrand. You must first evaluate the limit in the integrand. You must first evaluate the integral in the integrand. mult num and denom by 1-cos x mult num and denom by 1+cos x mult num and denom by 1-sin x mult num and denom by 1+sin x Argument of tan must be the variable of integration. Exponent of csc must be an integer. reduce $\int csc^n t dt$ Swap selected row with which row? Multiply selected equation by what? Divide selected equation by what? Add ? times selected equation to equation ? Subtract ? times selected equation from equation ? Solve selected equation for which variable? Multiply selected row by what? Divide selected row by what? Add ? times selected row to row ? Subtract ? times selected row from row ? Coefficient must be non-negative. Denominator must be positive. a/c must be positive. Too many subscripted variables, can't make more. MathXpert would take one more step. MathXpert would take two or more additional steps. Close this calculation and select or enter another problem? Equation is not linear in the selected variable. solve for if ln u=v then u=e^v if log u=v then u=10^v if log_b u=v then u=b^v Change the base of logarithms first. That operation ran out of memory. Sorry, try something else. Your graph is being prepared for printing. Please Wait Printer not found Not enough memory to print. Number incorrectly entered. Start the next problem Start the previous problem Gives mathematical advice Confirm the problem is finished Make a relevant graph MathXpert will finish the problem MathXpert will take one step Undo the last step Double vertical range (arrow keys work too) Halve vertical range (arrow keys work too) Double horizontal range (arrow keys work too) Halve horizontal range (arrow keys work too) Increase parameter value (plus key works too) Decrease parameter value (minus key works too) Set parameter value or increment Point and Slope Select new drawing region Select centered drawing region Move the graph with the mouse. If you have a touch screen or trackpad, you can move the graph with two fingers. Move the Spot Parameter value corresponding to the Spot Redraw, using selected rectangle No use dividing by 1, nothing will change. Expression must be in polynomial form. You must select the expression to be worked on. Leading coefficient is already 1. make leading coef 1 You entered this problem under integration by substitution, but it may need integration by parts. Cannot calculate the limit of the base. Cannot calculate the limit of the exponent That would create 0 to the power 0, which is undefined. That would create 0 to an undefined power, which is undefined. However, that doesn't prove your limit is undefined. That would create 1 raised to an undefined or infinite power. The limit cannot be evaluated this way. That would create infinity raised to the power 0. Limit of exponent is undefined. Dividing by a quantity containing the variable is allowed only if the quantity can never be zero; otherwise you might lose some solutions. Ensure circles appear circular Show or hide the direction field Can't compute roots numerically. MathXpert is already running. MathXpert will not multiply out terms with more than 64 summands. First get the indexed sum outside the integral. Finished Problem leading term Can't use leading terms because exponent isn't constant. In $^n\sqrt a^m$, a must be negative. It's already written as a function of that expression. Solution terminated due to excessive length. You were working on a saved calculation, so 'next problem' does not make sense. You were working on a saved calculation, so 'previous problem' does not make sense. Beautiful (Gouraud shading) Fast (transparent wire frame) Opaque wire frame eqn identically true To get a meaningful answer, you must specify values of the variables. Select the whole expression to do so. Type it in Left Middle Right Enter your substitution like this: x = sin u Cannot infer that new root(s) would be defined. evaluate numerically Error $a < x^(2n)$ is true if $a < 0$ $a \le x^(2n)$ is true if $a \le 0$ $x^(2n) < a$ is false if $a \le 0$ $x^(2n) \le a$ is false if $a < 0$ Use the mouse to select an expression you wish to change. Choose the kind of problem you want to solve or graph you want to draw. Select or enter the problem to be solved. Select or enter the function to be graphed. Select or enter the functions to be graphed. factor integer x in $\sqrt x$ factor integer x in $^n\sqrt x$ See how Auto Step would proceed absorb constant combine intervals subexpression not everywhere defined. Remember !Solutions may be lost due to new assumptions made at this step. Right side must be non-negative. The two sides must have the same sign. Too difficult. You can still try graphical solution. Divide by the nonzero common factor, instead of using this operation. square both sides Steps have been taken that would require checking the solution. That would mean verifying the original equation as an identity. MathXpert is unable to do that, so the operation fails. !This assumption was made earlier. !There are more assumptions that may restrict the solution further. Left side must be non-negative. Right side must be positive. You may gain unwanted solutions by eliminating a Multiply both sides by the square of what? Discriminant must be non-negative. Discriminant must be negative. Cannot determine the sign of the linear term. Cannot determine the sign of the cubic term. Result $a$ must be constant. $a$ must be positive. $a$ must be non-negative. $b$ must be positive. $b$ must be constant. use assumptions Cannot determine the sign of the expression you chose. Right side is negative. Left side is negative. substitute Equation must be a cubic with no quadratic term. !Solutions may be lost if the substitution does not take all possible values. Right side of equation must be zero. Equations must be solved for the new variable. eliminate reject unsolvable eqn This equation may or may not have solutions, depending on the values of the parameters. Enter the exponent n polar form simplify Numerator can't be 0 Exponent cannot exceed 100 That substitution would require making an assumption, which might cause you to lose some solutions. Since the common factor of the two sides can't be zero in this case, you should just divide by it. That integer has factors, but not small ones, and it would take a long time to search for them. That integer has no factors less than four billion. Left side must be negative. Left side must not be positive. Left side must be non-negative v>|u| iff -v < u < v $v\ge |u|$ iff $-v \le u \le v$ |v|>u iff v<-u or v>u $|v|\ge u$ if $v\le -u$ or $v\ge u$ $u>v$ or u=v iff $u\ge v$ -u < -v iff u > v -u > -v iff u < v $-u \le -v$ iff $u \ge v$ $-u \ge -v$ iff $u \le v$ $^(2n)\sqrt u<v$ iff $u<v^(2n)$ (u>0) $v>^(2n)\sqrt u$ iff $v^(2n)>u (u>0)$ $^(2n)\sqrt u\le v$ iff $u\le v^(2n)$ (u>0) $v\ge ^(2n)\sqrt u$ iff $v^(2n)\ge u$ (u>0) $^(2n)\sqrt u<v$ iff $0\le u<v^(2n)$ $v>^(2n)\sqrt u$ iff $0\le u<v^(2n)$ $^(2n)\sqrt u\le v$ iff $0\le u\le v^(2n)$ $v\ge ^(2n)\sqrt u$ iff $0\le u\le v^(2n)$ Left side must be positive. take reciprocals Right side must be negative. u/v > 0 => v>0 if u>0 $u/v \ge 0$ => $v\ge 0$ if $u\ge 0$ $u/v \ge 0 => u\ge 0$ if v>0 $u/v > 0 => u>0$ if v>0 u<v iff ln u < ln v u<v iff log u < log v $u\le v$ iff $ln u \le ln v$ $u\ge v$ iff $ln u \ge ln v$ u < v iff a^u < a^v u > v iff a^u > a^v $u \le v$ iff $a^u \le a^v$ $u \ge v$ iff $a^u \ge a^v$ u < ln v iff e^u < v ln u < v iff u < e^v $u \le ln v$ iff $e^u \le v$ $ln u \ge v$ iff $u \ge e^v$ ln u > v iff u > e^v u > ln v iff e^u > v $ln u \le v$ iff $u \le e^v$ $u \ge ln v$ iff $e^u \ge v$ u < log v iff 10^u < v log u > v iff u > 10^v log u < v iff u < 10^v u > log v iff 10^u > v $u \le log v$ iff $10^u \le v$ $log u \ge v$ iff $u \ge 10^v$ $log u \le v$ iff $u \le 10^v$ $u \ge log v$ iff $10^u \ge v$ Enter $a$ in $a^u \le a^v$ ? Enter $a$ in $a^u > a^v$ ? Enter $a$ in$ a^u \ge a^v$ ? Don't ask this question in the future. If you turn off confirmation here, you can turn it back on from the Options menu. Select a new problem (or edit this one) u>v>0 iff ln u > ln v u>v>0 iff log u > log v $cot m\pi $ is undefined sin u = c iff u=arcsin(c)+2n$\pi $ or $u=2n\pi +\pi -arcsin(c)$ Equations must first be solved. Original equation is not periodic. Solution does not contain an integer parameter. periodic form You were working on a saved graph, so 'previous problem' does not make sense. Close this graph and draw the next one? Close this graph and draw the previous one? Draw the next graph Draw the previous graph Select a new function (or edit this one) Change the ranges of the variables Not yet solved. There is an unevaluated limit term. There is an unevaluated indexed sum. Identity not yet verified. The two sides are not identical. ✅ That's a good answer. ✅ That's the answer: the identity is verified. There is still an unevaluated derivative. There are still fractions to put over a common denominator. There is still a compound fraction to be eliminated. There is still a negative exponent to be eliminated. There is still a root to be eliminated. There is a product of sums to multiply out. There is a power of a sum to expand. Not yet Your answer is acceptable, but perhaps it could be simplified. MathXpert would take one more step. Your answer is acceptable, but perhaps it could be simplified. MathXpert would take two more steps. Your answer is acceptable, but perhaps it could be simplified. MathXpert would take at least three more steps. You have included solutions that are ruled out by current assumptions. Use the assumptions to correct the solution. Cannot eliminate the possibility that the base is zero. Cannot eliminate the possibility that the base is one. Equation is an identity, true whenever both sides are defined. Change the graph paper showing in the button Green means erase old curves when the parameters change. Red means keep them. $-c\le |u|$ is true ($c\ge 0$) -c<|u| is true (c>0) $|u|<-c$ has no solution if $c\ge 0$ $|u|\le -c$ has no solution if c>0 The right-hand side must be non-positive. Right-hand side can never be zero. $|u| \le -c$ iff u = 0 (assuming c = 0) 0>|u| has no solution -c > |u| has no solution ($c \ge 0$) $-c \ge |u|$ has no solution (c > 0) The left-hand side must be non-positive. Left-hand side can never be zero. $-c \ge |u|$ iff u = 0, (assuming c = 0) $|u|\ge -c$ is true ($c\ge 0$) |u|>-c is true (c>0) !Only one solution satisfies the original inequality for !Keep only solutions that satisfy the original inequality. swap rows User |u| = -c iff u = 0 (assuming c = 0) apply function Enter the name of a function, for example 'log' The sign of the discriminant is not determined. This looks solved, but the variable is not the original one. This is solved, but MathXpert would simplify the answer further. You are not allowed to multiply by zero. That would make the equation true for all values of the unknown. You have taken some steps (for example squaring or 'simplify') that might have introduced new solutions. Check your solutions. You are not allowed to multiply by an expression containing the unknown. The resulting equation would no longer be linear. That would make the equation true for all values of the unknowns. Cannot verify that the chosen expression is nonzero. If it has zeroes that are not zeroes of the inequality, the solution set would change. Therefore this multiplication is illegal. evaluate to rational Specify details such as orientation or number of copies. Print the active calculation or graph immediately. Displays version information and credits. Conditions under which the calculation is valid. Functions can be used until they are undefined. Replace an existing definition with a new one. Delete a function definition. Display the currently valid function definitions. Change calculation background color. The normal color of displayed formulas The highlight color is used to show where changes took place. The color to be used for the justifications Restore the original colors, as before they were ever changed. Use a larger font size. Use a smaller font size. Restore the original font size. The buttons at the top of a calculation will be large. The buttons at the top of a calculation will be small. This option affects display only, not internal accuracy. Use commas when displaying large integers. Do not separate the digits of large integers. You will hear applause when you complete a problem yourself. Operate silently. Close problem file Integrate by parts with Creates a new menu item for direct access to mathematical operations. Remove the Operations menu. Arrange several graphs and calculations like a hand of cards. Make all graphs and calculations visible at once. Affects graphs and calculations that have been minimized. Close all open graphs and calculations. Change the domain and range intervals. Change the current values or increments of parameters. Place numbered marks (ticks) on the axes. Do not place numbered marks (ticks) on the axes. Place ticks at multiples of $\pi$ on the horizontal axis. Use this with No Ticks to retain some indication of scale. Label the axes as usual with variable names. Do not label the axes in any way. Specify initial values when solving differential equations Specify spacing between contour lines. Specify the coordinates of the virtual camera. Specify the coordinates of the virtual lamps. Singularities are places where the function has an infinite limit. Jumps are places where the two one-sided limits are different. Controls whether titles are displayed on graphs or not. A grid is a background pattern, similar to graph paper. Hiding the Toolbar buttons does not affect printing. The color used for formulas selected with the mouse. Store a selected term for later pasting into another graph or calculation. Explains how to use the Help system. Enters the MathXpert help system. Search for help on a specific topic. Remark No printer installed. Unknown error. Perhaps the printer is incorrectly installed. It is already a polynomial in that expression. Can't determine the sign of the expression by which you wanted to divide. Can't determine the sign of the expression by which you wanted to multiply. To ensure that the result is defined, you would have to make assumptions involving the variable. That might cause you to lose solutions, and hence is not allowed. A common numerical factor has been removed first. Now you can try to factor the remaining quadratic. The function you attempted to graph is nowhere defined. to use the assumptions to eliminate such unwanted solutions at the end. !An assumption has been made to ensure that the discriminant is non-negative. !Assuming to ensure that the discriminant is non-negative. !Assuming the left side is non-negative. !Assuming the right side is non-negative. !Assuming the two sides have the same sign. Coordinates of selected point That nonnegative expression is already in polar form. That expression is not in polar form. Polar form is $re^i^(\theta).$ Reduce complex numbers to the form $a + bi$. Denominator is not real Substitution illegal because it contains the summation variable. Substitution illegal because it contains the product variable. Substitution illegal because it contains the variable of integration. Substitution illegal because it contains the variable of differentiation. Substitution illegal because it contains the limit variable. Substitution illegal because it contains a bound variable. Enter the desired new exponent. substitute $u$,$v$... That expression still contains an inverse trig function. Cannot verify that $tan x$ is nonzero. Cannot verify that $cot x$ is nonzero. Power requested is undefined When solving linear equations, you can divide only by a constant. When solving linear equations, you can multiply only by a constant. You might lose solutions by applying that law without first verifying that condition. write integer as $a^n$ You can select one (or all) solutions and choose 'put solutions in periodic form'. You can't divide by that quantity, because MathXpert can't verify that it is never zero. In that case you might lose some solutions. The difference is very small, and possibly roundoff errors may have entered in, so the correctness of the calculation cannot be guaranteed. write as square write as cube express in form a^? Assuming the new square roots are defined. The resulting expression is equal to the previous one only subject to those assumptions. For example, its graph may be different. You can only convert a specific, numerical angle to degrees. Definition of a new variable cannot be constant. The group entered must be a sum. $a\sqrt b = \sqrt (a^2b)$ if $a\ge 0$ The factor outside the square root must be nonnegative The factor outside the root must be nonnegative $$a root(n,b) = root(n,a^n b)$$ (n odd) $$a root(n,b) = root(n,a^n b)$$ ($a\ge 0$) Try again, choosing letters in the last half of the alphabet for your variables, and letters in first half (or Greek letters) for your constants. Inappropriate problem Too many windows You have solved the equations, but perhaps the answers should be simplified. You have solved the equation, but MathXpert would take one more step. You have solved the equation, but perhaps the answer should be simplified. Because the topic is Trig Addition, a solution using a trig addition formula will be given, instead of a shorter solution using periodicity. Although a formula for the singularities has been calculated, it was too hard for MathXpert to find their numerical values. MathXpert calculation MathXpert graph Please enter a positive integer. Zero is not positive. $-^n\sqrt a = ^n\sqrt (-a)$ if n is odd This equation has more than one real root, and its solution requires complex numbers, which are not used in this topic. There can be at most three roots of a cubic equation. Simplify the expressions for the roots until some of them are identical. rationalize denom and simplify Enter the new exponent: You have solved the equation, but the answer should still be simplified. Complex numbers are not in use. This law is not valid for real roots. Too many solutions will result. substitute specific integers a=-b becomes $a^2=-b^2$ if $a,b \ge 0$ a=-b becomes a=0 if $a,b \ge 0$ a=-b becomes b=0 if $a,b \ge 0$ A function with that same name has already been defined. You can redefine it or undefine it, but not define it. $lim(t\to \pm \infty ,cos t)$ is undefined $lim(t\to \pm \infty ,sin t)$ is undefined $lim(t\to \pm \infty ,tan t)$ is undefined Limit of the denominator is not defined and not plus or minus infinity. Limit of the numerator is not defined and not plus or minus infinity. $a/^n\root b = ^n\root (a^n/b)$ (n odd or $a\ge 0$) $^n\root a/b = ^n\root (a/b^n)$ (n odd or $b>0$) $(\sqrt a)/b = \sqrt (a/b^2)$ if $b>0$ $a/\sqrt b = \sqrt (a^2/b)$ if $a\ge 0$ Numerator must be non-negative Denominator must be non-negative Enter the first new exponent: First evaluate the derivative. No equation is solved for a derivative of a variable. eliminate derivative You have already differentiated the equation. You must solve each equation, either for a derivative or a variable. Limit of denominator is undefined. Is This law is not generally valid; what is under the root must be real. What is under the root must be negative. sgn(x) = 1 if x > 0 sgn(x) = -1 if x < 0 sgn(x)=|x|/x (x nonzero) sgn(x)=x/|x| (x nonzero) d/dx sgn(u) = 0 (u nonzero) $\int sgn(u)v dx=sgn(u)\int v dx$ (u nonzero) Expression inside sgn may be zero. The expression inside the sgn function cannot be positive. sgn(x)=1 assuming x>0 The expression inside the sgn function cannot be negative. sgn(x)=-1 assuming x<0 Enter n = integer - k, where n is the variable to change. sgn(ax) = sgn(x) if a>0 sgn(ax) = -sgn(x) if a<0 sgn(ax/b)=sgn(x), a/b>0 sgn(ax/b)=-sgn(x),a/b<0 MathXpert cannot calculate that limit. The resulting numerator and denominator would both be undefined. The resulting numerator and denominator would both be zero. The resulting denominator would be zero, and MathXpert can't determine its sign. Multiply one factor and divide the other by what? One limit would be zero and the other undefined. That expression is real, but it is negative, so it's not in polar form, which is $re^(it)$ with $r$ positive. That expression still contains a product of trig functions. Initial value of function Initial value of derivative Initial value of second derivative Initial value of third derivative Initial value of fourth derivative First evaluate the limit. $lim a/u^2 = \infty $ if $lim u = 0$ and $a > 0$ $lim a/u^2 = -\infty $ if $lim u = 0$ and $a < 0$ $lim a/u^2^n = \infty $ if $lim u = 0$ and $a > 0$ $lim a/u^2^n =-\infty $ if $lim u = 0$ and $a < 0$ $lim u^n = (lim u)^n$ requires $lim u$ to be defined, infinite, or oscillatory, MathXpert cannot verify either condition. That is not the case here. Eliminate zero denominators. or when $u$ is nonnegative or nonpositive. Evaluate to apply the divergence test $lim uv = lim u lim v$ can fail if $lim u = 0$ and $lim v$ is undefined. Enter one term of the new sum: This inequality is too complicated for MathXpert to graph. $a < \sqrt u$ iff $0 \le u$ provided a < 0 $a \le \sqrt u$ iff $0 \le u$ provided $a \le 0$ $\sqrt u > a$ iff $u \ge 0$ provided a < 0 $\sqrt u \ge a$ iff $u \ge 0$ provided $a \le 0$ $a < ^2^n\sqrt u$ iff $0 \le u$ provided a < 0 $a \le ^2^n\sqrt u$ iff $0 \le u$ provided $a \le 0$ $^2^n\sqrt u > a$ iff $u \ge 0$ provided a < 0 $^2^n\sqrt u \ge a$ iff $u \ge 0$ provided $a \le 0$ One of the factors is zero. One of the factors is, or simplifies to, zero. If not, you may get a wrong answer, so be careful. MathXpert will assume it is defined and nonzero. MathXpert is assuming that what is under the square root is nonnegative. It is possible to make contradictory assumptions and arrive at wrong answers. Therefore, keep an eye on your assumptions. Base of exponent must be positive. There is an unevaluated infinite or undefined subterm. Denominator too complicated to expand in partial fractions. you must first prepare for it by using Before you are ready to use You might try !This equation can be solved for the original integral. !provided this limit turns out to be zero. Input not acceptable $(e^(-t)-1)/t\to $ $(sinh t)/t\to $ $(tanh t)/t\to $ $(cosh t-1)/t\to $ $(1-cosh t)/t\to $ $(cosh x-1)/x^2\to \onehalf $ as $x\to $ Disabled Restricted Trial The problem file for this topic has been altered. Restore the original problem files or reinstall MathXpert. from Previous Next Help Edit Paste Problem Number Source Equations valid for all times: Equations valid for only one time: Graph Number Domain and Range Style Riemann Sums From to Number of Intervals Style of Sum Simpson's Rule Trapezoid Rule Is this what you meant? Original Current Slope Selected Point Active is now with increment After parameter changes Erase old graphs Show all graphs Parameters Initial Values After changing initial values Graph Colors Color Border Function Background Axes with Title is Ranges Horizontal Vertical First Graph Second Third Fourth Fifth Sixth Restore Original Parametric Graph Ranges Parameter Ensure circles are round Polar Graph Ranges Angle Textbook Selection Select a textbook About MathXpert Design and implementation: Michael Beeson Lab Edition Book dialog: Dave Empey Translation: Splash screen art: Cindi Shih Copyright © 2004 Help With Math Please wait Print Left Header Right Header Size and Position of Graph Inches Centimeters Left Margin Top Margin Don't ask this question in the future Exit Save Save as Print now Define a function Redefine a function Undefine a function Show functions Always request confirmation Never request confirmation Formulas Highlighted Formulas Selected Formulas Justifications Restore Factory Colors Type Size Bigger Smaller Normal Tool Bar Large Icons Small Icons Decimals Large Numbers Sound On Off Confirm Close Language Operations Show Operations Menu Hide Operations Menu Cascade Tile Arrange Icons Close All Contents Search for Help on... Using help Ticks None Multiples of $\pi$ Axes Labels Numbers Letters No Labels Change Colors All Graphs Approximate Integration Differential Equations $$root(n,-a) = -root(n,a)$$ ( $n$ odd) $a<u^2<b$ => $-\sqrt b<u<-\sqrt a$ or $\sqrt a<u<\sqrt b$ $a\le u^2\le b$ => $-\sqrt b\le u\le -\sqrt a$ or $\sqrt a\le u\le \sqrt b$ $a<u^2\le b$ => $-\sqrt b\le u<-\sqrt a$ or $\sqrt a<u\le \sqrt b$ $a\le u^2<b$ => $-\sqrt b<u\le -\sqrt a$ or $\sqrt a\le u<\sqrt b$ $a < u^2^n$ iff $u<-^2^n\sqrt a$ or $^2^n\sqrt a<u$ $a\le u^2$ iff $u\le -\sqrt a$ or $\sqrt a\le u$ $a<u^2^n<b$ => $-^2^n\sqrt b<u<-^2^n\sqrt a$ or $^2^n\sqrt a<u<^2^n\sqrt b$ $a\le u^2^n\le b$ => $-^2^n\sqrt b\le u\le -^2^n\sqrt a$ or $^2^n\sqrt a\le u\le ^2^n\sqrt b$ $a<u^2^n\le b$ => $-^2^n\sqrt b\le u<-^2^n\sqrt a$ or $^2^n\sqrt a<u\le ^2^n\sqrt b$ $a\le u^2^n<b$ iff $-^2^n\sqrt b<u\le -^2^n\sqrt a$ or $^2^n\sqrt a\le u<^2^n\sqrt b$ $\sqrt (a^2^n)=a^n$ if $a^n\ge 0$ $|sin u| \le u$ if $u\ge 0$ $|arctan u| \le u$ if $u\ge 0$ $|tan u| \le u$ if $0\le u\le \pi /2$ $$lim(x->a,c)= c$$ (c constant) $lim c^v = c^(\lim v)$ compute factorial shift sum limits $\sum cu = c\sum u$ (c const) defn of $d^nu/dx^n$ differentiate matrix for power series $\sqrt a/b = \sqrt (a/b^2)$ if b>0 $\sqrt a/b= -\sqrt (a/b^2)$ if b<0 (b>0 or n odd) (b<0, n even) if $a\le 0$ (n odd or $a\ge 0$) (n even and $a\le 0$) if $-\pi /2\le \theta \le \pi /2$ if $0\le \theta \le \pi $ is undefined $\sqrt (-a) = i\sqrt a$ if $a\ge 0$ Number inside ln must be in polar form. Number inside $\sqrt $ must be in polar form. Number inside $^n\sqrt $ must be in polar form. Use 'clear denom of i' instead [p=a,p=-a] and $p\ge 0$ iff p=|a| |u| = u iff $0 \le u$ |u| = -u iff $u \le 0$ $0\le u/v$ => 0<uv or u=0 $u/v\le 0$ => uv<0 or u=0 (if n is odd) $u^2<a iff -\sqrt a < u < \sqrt a$ $a<u^2 iff u<-\sqrt a or \sqrt a<u$ $a\le u^2 iff u\le -\sqrt a or \sqrt a\le u$ $-a<x^2<b$ iff $x^2<b$ $-a\le x^2\le b$ iff $x^2\le b$ $-a<x^2\le b$ iff $x^2\le b$ $-a\le x^2<b$ iff $x^2<b$ $u^2^n < a$ iff $-^2^n\sqrt a < u < ^2^n\sqrt a$ $u^2^n \le a$ iff $-^2^n\sqrt a \le u \le ^2^n\sqrt a$ $u^2\le a iff -\sqrt a \le u \le \sqrt a$ cancel cancel gcd Divide row by Add times row to row Sub from row matrix multiplication formula for 2 by 2 matrix inverse convert matrix eqn where cos in $a|b| = |ab|$ if $0 \le a$ a|b|/c = |ab/c| if $0\le a/c$ Completing the square requires a non-constant linear term. This equation is solved, but it is customary to put the variable on the left side, so MathXpert would take one more step. !Assuming the previous right side is nonnegative. Find the extrema on You must compute a table of function values and select the maximum or minimum. You should select the maximum and/or the minimum to complete the problem. !No minimum or maximum on this interval Find the extrema. This operation cannot be applied to an improper integral unless the integrand is nonnegative. There is an unevaluated infinite series. MathXpert cannot find a closed form for this series. The general term does have a finite limit. telescoping series MathXpert cannot compute the limit of the general term. Expression to shift by cannot depend on the index variable. (n odd) function is constant Max and min are the same for a constant function. It is not necessary, and will not help, to take the derivative. Perhaps you are trying to reject an endpoint. You can't insert more than 1000 new terms. add to index variable subtract from index variable How many terms do you want explicitly written out? Decrease the lower limit of summation by how much? Add what to the index variable? Subtract what from the index variable? decrease lower limit MathXpert cannot check the convergence of the series that would result. The resulting series would be divergent, so the law cannot be applied. The function being defined cannot be used in the definition. Evaluate to apply the integral test Evaluate to apply the root test Evaluate to apply the ratio test Evaluate to apply the comparison test The general term is not decreasing. MathXpert cannot determine that the general term is decreasing. !The series is convergent. !The series is divergent. Some assumptions have been made. !The test was inconclusive. finish integral test finish root test finish ratio test finish comparison test finish divergence test You must first evaluate the limit, or undo. The comparison term does not contain the summation variable. Evaluate to apply the limit comparison test finish limit comparison test The test worked; the problem is reduced to the convergence of this series. Cannot verify the required inequality Enter the general term of the comparison series The new integrals would not be defined, so this operation is illegal. The integrand is not defined on the entire interval of integration. You should break it into two or more integrals first. The integrand is undefined somewhere besides the left endpoint. The integrand is undefined somewhere besides the right endpoint. MathXpert cannot evaluate the limit. The limit is zero. limit of integrand is not zero Although there are singularities, the integrand is nonnegative, so the integral is infinite rather than just undefined. There are singularities. You might be able to get the answer $\infty $ or $-\infty $ if you first break the integral, unless it contains infinities of both signs. When both limits are infinite, you must break the integral first. The limits at the two ends must exist separately. limit of integrand is undefined condensation test finish condensation test !The test worked. The series is convergent. Assumptions may have been made. !The test worked. The series is convergent. !The test worked. The series is divergent. The limit of the general term is not zero. You started the divergence test, but didn't finish it yet. You started the integral test, but didn't finish it yet. You started the comparison test, but didn't finish it yet. You started the limit comparison test, but didn't finish it yet. You started the condensation test, but didn't finish it yet. You started the root test, but didn't finish it yet. You started the ratio test, but didn't finish it yet. The convergence or divergence is not yet settled. calculate first terms Maximum number of first terms to add is about 128,000. Lower index of sum cannot exceed sixteen million. Calculate the sum of the first how many terms? !If this series converges, so does the original series. !If this series diverges, so does the original series. The answer must be expressed in terms of the original variable or variables. is not correct. Cannot switch to that language. express series as $a_0 + a_1 + ...$ express series as $a_0 + a_1 + a_2 + ...$ express series using ... and general term show more terms before ... express series using sigma notation show another term before ... The maximum number of terms before ... is 1000. How many more terms do you want to see? solve for constant of integration The resulting series would not converge. The resulting series would converge only at isolated points integrate power series term by term differentiate term by term You must first rename one of the summation variables. multiply series multiply power series divide power series add series subtract series Lower limits of summation are not the same. Upper limits of summation are not the same. You must rename one or more of the summation variables first. binomial series First rewrite one or both of the series, to arrange that the exponents have the same coefficient of the summation variable. First term in the denominator must be the constant term. MathXpert requires numerical coefficients for series division. Constant term must be nonzero for series division. Too many function definitions. Undefine a function to make room for the coefficients of the quotient. MathXpert cannot verify that the recursion will terminate. Function argument must be an integer. Function argument too large. compute function value show terms with factorials evaluated show coefficients in decimal form do not use decimal form for coefficients do not evaluate factorials MathXpert requires numerical coefficients for series exponentiation. Exponent must be a number. express $(\sum a_k x^k)^n$ as a series Leading coefficient must be nonzero for series exponentiation. if otherwise express as power of ? Enter the desired base: Cannot draw the graph Function is not a non-constant polynomial for the current parameter value. Change the parameter value. Function is not a non-constant polynomial for the current parameter values. Change the parameter values. It is not correct to apply this operation to an improper integral. Find the minimum and maximum of on the interval Sorry, the term you entered is not a factor. The law $ln ab = ln a + ln b$ is not always correct. The two sides can differ by a multiple of $2\pi i$ . For example, try the case when $a$ and $b$ are both $-1$. !Assuming that the radius is positive. Range Redraw Prev Undo AutoStep ShowStep AutoFinish Graph UHSWAFGPEN graphs Integration by parts cannot be applied to improper integrals directly. First express the integral as a limit. That function is still in use. To redefine it you must first close all calculations and graphs that mention it. Can't verify exponent is even. Exponent is not even. Solution(s) should be reduced to the form $a + bi$. Division by zero is not allowed. That would produce the useless equation 1 = 1, since any nonzero number to the zero-th power is 1. That would produce the useless equation 0 = 0, since zero raised to any nonzero power is zero. That would produce a fraction with zero in the denominator, which is not defined. That expression is already in the form $u+iv$. Expression should be reduced to the form $a + bi$. result of comparison test You still need to state the final result of the comparison test for the original series. Red Gray Blue Green Cyan Magenta Yellow White Black thin thick no ticks normal ticks ticks at $n \pi$ Please use a color different from the background. and $-a \le u \le a$ iff $|u|\le a$ $-a < u < a$ iff $|u|<a$ Select Graph Paper Adjust to Graph Paper Automatically adjust colors Title Title Background Light blue Medium blue Light orange Light green Medium gray Light gray Light yellow Light violet Apply changes to Graph 1 Graph 2 Graph 3 Graph 4 Graph 5 Graph 6 Please use a title color different from the background. Please use an axes color different from the background. Please use a border color different from the background. Please use a function color different from the background. Please use function colors different from the background. Width Height Landscape Portrait Whole page, with specified margins Maximum size Same shape as on screen Center vertically Center horizontally Save settings without printing Left margin plus graph width cannot exceed paper width. Top margin plus graph height cannot exceed paper height. Graph width must be positive. Graph height must be positive. tan(t/2) is undefined somewhere in the interval of integration. MathXpert cannot verify that tan(t/2) is defined on the interval of integration. Negative real numbers have no complex square roots. $(a^(it))^c = a^(it)^c$ only when $-\pi < t \le \pi$ First write a in polar form. Equality sign expected. $\sqrt(bi)= \sqrt(b/2)+\sqrt(b/2)i$ if b >= 0 $\sqrt(-bi)= \sqrt(b/2)-\sqrt(b/2)i$ if b >= 0 $\sqrt(a+bi)= \sqrt((a+c)/2)+\sqrt((a-c)/2)i$ if b >= 0 and c=\sqrt(a^2+b^2) $\sqrt(a-bi)= \sqrt((a+c)/2)-\sqrt((a-c)/2)i$ if b >= 0 and c=\sqrt(a^2+b^2) There is a repeated variable on the left side. ac=bc iff a=b or c=0 The selected expression must contain the variable of integration. Make a Graph Work a Problem Run a Demo Point, Slope, Parameter Radians Degrees Go to Help With Math's web site Send email to Help With Math Author's essay about MathXpert Switch MathXpert to the specified language Double vertical range Halve vertical range Double horizontal range Halve horizontal range Enter the new lower limit of summation: We keep only one of the two solutions, since each of the two will generate the same solutions of the cubic equation. Grapher You are not currently enrolled in any course using WebGrades. evaluate Bernoulli number exactly evaluate Euler number exactly MathXpert cannot decide whether this series converges or not. The series must be absolutely convergent, and MathXpert cannot verify that it is. !We will try to prove the series converges. !We will try to prove the series diverges. � , F ? ? � S L 6.
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