Calculus Homework Answers & Questions

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  • Waiting for answer How do you find the derivative of ##y=cos(x)## from first p...

    Using the definition of a derivative: ##dy/dx = lim_(h- 0) (f(x+h)-f(x))/h##, where ##h = deltax## We substitute in our function to get: ##lim_(h- 0) (cos(x+h)-cos(x))/h## Using the Trig identity: ##cos(a+b) = cosacosb - sinasinb##, we get: ##lim_(h- 0) ((cosxcos h - sinxsin h)-cosx)/h...

  • Waiting for answer How do you use the epsilon delta definition to prove that th...

    Please see below. The preliminary analysis is a bit long. If you just want to read the proof, scroll down. Preliminary analysis We want to show that ##lim_(xrarr2)(x^2-7x+3) = -7...

  • Waiting for answer What is a corner in calculus?

    A derivative at a specified point is only defined for a function where there is only one slope at that specified point. A corner is one type of shape to a graph that has a different slope on either side. It is similar to a cusp . You may see corners in the context of absolute value functions, like...

  • Waiting for answer How do you integrate ##ln(x^2-x+2)dx##?

    ##=(x- 1/2) ln(x^2-x+2) - 2x + sqrt 7 arctan ( (2x-1)/sqrt 7)## ##int dx qquad ln(x^2-x+2)## we use IBP ##int u v' = uv - int u' v## and the trick here is ##u = ln(x^2-x+2),...

  • Waiting for answer If a 200 pound cable is 100 feet long and hangs vertically f...

    Since the density of the cable is ##{200}/{100}=2## lb/ft, the small piece of the cable of length ##dx## pulls down with the force ##2dx## lb. Let ##x## be the distance of the piece of the cable from the top of the building. To lift this piece up to the top, it requires the force of ##dW=2d...

  • Waiting for answer Find the volume of the largest rectangular box in the first...

    By solving for ##z##, ##x+2y+3z=6 Leftrightarrow z= 2-1/3x-2/3y## Let ##(x,y,z)## be the vertex on the plane, where ##x,y,z 0##. The volume ##V## of the rectangular box can be expressed as ##V=xyz=xy(2-1/3x-2/3y)=2xy-1/3x^2y-2/3xy^2## Second Partial Test Let us find critical points. ##V_x=2...

  • Waiting for answer What is the derivative of ##sinx(sinx+cosx)##?

    The answer is ##cos^2(x)-sin^2(x)+2cos(x)sin(x)=cos(2x)+sin(2x)## First, use the to say ##d/dx(sin(x)(sin(x)+cos(x)))=cos(x)(sin(x)+cos(x))+sin(x)(cos(x)-sin(x))## Next, expand t...

  • Waiting for answer What is the antiderivative of ##3e^x##?

    ##3e^x+C## You should already know that the derivative of ##e^x## is just ##e^x##. Also, when differentiating, multiplicative constants remain and are not altered. Since the two com...

  • Waiting for answer Suppose that we don't have a formula for g(x) but we know th...

    Linearization of f(x) at a ##L(x)=f(a)+f'(a)(x-a)## Let us find the linearization ##L(x)## of ##g(x)## at ##1##. Since ##g(1)=-3## and ##g'(1)=sqrt{(1)^2+8}=3##, ##L(x)=g(1)+g'(1)(x-1)=-3+3(x-1)=3x-6##. So, we can approximate: ##g(0.9) approx L(0.9)=3(0.9)-6=-3.3## ##g(1.1) approx L(1.1)=3(...

  • Waiting for answer What is the line of symmetry of the graph of ##y=1/(x-1)##?

    The graph is a hyperbola, so there are two lines of symmetry: ##y=x-1## and ##y=-x+1## The graph of ##y = 1/(x-1)## is a hyperbola. Hyperbolas have two lines of symmetry. both li...

  • 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...

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