Calculus Homework Answers & Questions

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  • Waiting for answer How do you remove a removable discontinuity?

    You can remove a removable discontinuity by redefining the function at the "problem spot". Removable discontinuities are where the graph of a function has a hole. This will occur in rational functions at an x-value that makes both the denominator and the numerator equal zero. For example, the grap...

  • Waiting for answer How do you know when to use L'hospital's rule twice?

    L'hospital's rule is used when an initial evaluation of a limit results in an indeterminate form such as ##0/0## or ##(+-oo)/(+-oo)## If, after application of L'hospital's rule, your limit evaluation produces another indeterminate form you apply L'ospital's rule again. L'hospital's rule st...

  • Waiting for answer What is the derivative of ##y=ln(sec(x)+tan(x))##?

    Answer: ##y'=sec(x)## Full explanation: Suppose, ##y=ln(f(x))## Using , ##y'=1/f(x)*f'(x)## Similarly, if we follow for the problem, then ##y'=1/(sec(x)+tan(x))*(sec(x)+tan(x))'## ##y'=1/(sec(x)+tan(x))*(sec(x)tan(x)+sec^2(x)) ## ##y'=1/(sec(x)+tan(x))*sec(x)(sec(x)+tan(x))## ##y'=sec(x)##...

  • Waiting for answer How do you use the limit definition to compute the derivativ...

    The definition of derivative in a generic point ##(x,f(x))## is: ##lim_(hrarr0)(f(x+h)-f(x))/h=f'(x)##. So: ##lim_(hrarr0)(cos(3(x+h))-cos3x)/h=lim_(hrarr0)(cos(3x+3h)-cos3x)/h=## than, using the formula sum-to-product : ##costheta-cosphi=-2sin((theta+phi)/2)sin((theta-phi)/2)##, ##=lim_(hrarr...

  • Waiting for answer What is the third derivative of tan x?

    Third derivative would be 2[##2sec^2 x tan^2 x +sec^4x]## If y= tan x y' = ##sec^2 x## y''= 2secx secx tanx= ##2sec^2 xtanx## y ''' = 2##[2sec x sec x tan x tan x+ sec^2 x sec^2 x}## = 2[##2sec^2 x tan^2 x +sec^4x]##

  • Waiting for answer How do you integrate ##int sin^2(x) dx## using integration...

    ##intsin^2(x)dx=x/2-sin(2x)/4+c##, where ##c## is the constant of integration. ##intsin^2(x)dx=intfrac{d}{dx}(x)sin^2(x)dx## ##=xsin^2(x)-intxfrac{d}{dx}(sin^2(x))dx## ##=xsin^2(x)...

  • Waiting for answer How do you integrate ##[(e^(2x))sinx]dx##?

    Integrate by parts twice using ##u = e^(2x)## both times. After the second , you'll have ##int e^(2x)sinx dx = -e^(2x)cosx + 2e^(2x)sinx - 4 int e^(2x)sinx dx## Note that the last...

  • Waiting for answer How do you find horizontal asymptotes for ##f(x) = arctan(x)...

    By definition, ##arctan x## is the inverse function of the restriction of the tangent function ##tan## to the interval ##(-pi/2,pi/2)## (see ). The tangent function has vertical asymptotes ##x=-pi/2## and ##x=pi/2##, for ##tan x=sin x/cos x## and ##cos \pm pi/2=0##. Moreover, the graph of the inv...

  • Waiting for answer What is the Derivative of root x + 1/root x?

    See explanation. If you mean: ##sqrt(x)+1/sqrt(x)## Derivative is given by: ##1/(2sqrt(x)) - 1/(2sqrt(x^3))= (x-1) / (2x^(3/2))## Wolfram Computational Engine If you mean: ##sq...

  • Waiting for answer What is Integration Using Simpson's Rule?

    Simpson's Rule will give you a better approximation of the integral than the other basic methods. The other methods are Rectangular Approximation Method (RAM) - left, middle, and right; and the Trapezoidal Rule. Numerical integration is used when we are given a set of data (evenly spaced on the in...

  • Waiting for answer What is the limit as x approaches 0 from the right-hand side...

    By observing the graph of ##y=ln x## below: we have ##lim_{x to 0^+}ln x=-infty## I hope that this was helpful.

  • Waiting for answer How many local extrema can a cubic function have?

    Simple answer: it's always either zero or two. Extrema are all maximum and minimum values of a function. Points of inflexion and deflexion are not extrema as you will see with the case of ##y=x^3## In general, any polynomial function of degree ##n## has at most ##n-1## local extrema, and polyno...

  • Waiting for answer How do you find the derivative of ## cos^2(2x)##?

    ##y=cos^2(2x)## ##dy/dx=2cos(2x)*-sin(2x)*2## ##dy/dx=-4cos(2x)*sin(2x)## ##dy/dx=-2sin(4x)##

  • Waiting for answer What is the derivative of ##((sinx)^2)/(1-cosx)##?

    ##y^' = -sinx## Here's an excellent example of how one simple trigonometric identity can spare you a significant amount of work on this derivative. More specifically, you can use th...

  • Waiting for answer What is the limit of ##( x^3 - 8 )/ (x-2)## as x approaches...

    Notice how you have a difference of two cubes. ##(x-y)(x^2 + xy + y^2) = x^3 + x^2y + xy^2 - x^2y - xy^2 - y^3 = x^3 - y^3## ##((x)^3 - (2)^3)/(x-2) = ((x-2)(x^2 + 2x + 4))/(x-2) = x^2 + 2x + 4## Plug in ##2##: ##lim_(x- 2)(x^3 - 8)/(x-2)= 4 + 4 + 4 = 12##

  • Waiting for answer How do I solve the initial-value problem: ##y'=sinx/siny;y(0...

    First of all, let's write ##y'## as ##dy/dx##. The expression becomes ##dy/dx = \sin(x)/\sin(y)## Multiply both sides for ##\sin(y) dx## and obtain ##\sin(y)\ dy = \sin(x)\ dx## integrating, one has ##\cos(y) = \cos(x)+ c## and thus ##y = \cos^{ -1}(\cos(x)+c)## This is the general solution...

  • Waiting for answer What is the Taylor Series expansion for f(x+2h)?

    Let 2h = a, ##f(x+2h) = f(x+a)## Taylor series of this is: ##f(x+a) = f(x) + f'(x)*a+ f''(x)*a^2/(2!) + f'''(x)*a^3/(3!) + ....## ##f(x+a) = sum_(n=0)^∞f^(n)(x)*a^n/(n!)## Substitute a = 2h, ##f(x+a) = sum_(n=0)^∞f^(n)(x)*(2h)^n/(n!)## To note : ##f^(n)(x)## means the ##n^(th)## differential of f(x...

  • Waiting for answer How do you find the derivative of ##sqrt(e^(2x) +e^(-2x))##?

    You would use the . ##y=(f(x))^n## ##y'= n(f(x))^(n-1)*f'(x)## ##y= sqrt(e^(2x) + e^(-2x))## write it in a form that is more understandable ##y = (e^(2x) + e^(-2x))^(1/2)## lets work out f'(x): remember: ##y = e^(ax)## ##y'(x) = ae^(ax)## ##f(x) = e^(2x) + e^(-2x)## ##f'(x) = 2e^(2x) + (-2)e^(...

  • Waiting for answer How do you find the derivative of ##sin2x - cos2x##?

    2cos2x + 2sin2x differentiate using##color(blue)" chain rule " ## ##d/dx [ f(g(x)) ] = f'(g(x)). g'(x)## apply this rule to each term ##d/dx(sin2x) = cos2x d/dx(2x) = 2cos2x ##...

  • Waiting for answer How do you find the integral of ##e^(7x)*sin(2x)dx##?

    By using twice. Let ##f(x)=e^(7x)## so that ##f'(x)=7e^(7x)##. Let ##g'(x)=sin(2x)## so that ##g(x)=-1/2cos(2x)##. Hence ##inte^(7x)sin(2x)dx=-1/2e^(7x)cos(2x)+7/2inte^(7x)cos(2x)dx## Now consider the integral ##inte^(7x)cos(2x)dx## Let ##f(x)=e^(7x)## so that ##f'(x)=7e^(7x)##. Let ##g'(x)=...

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