Calculus Homework Answers & Questions

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...
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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^27x+3) = 7...
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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...
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Waiting for answer How do you integrate ##ln(x^2x+2)dx##?
##=(x 1/2) ln(x^2x+2)  2x + sqrt 7 arctan ( (2x1)/sqrt 7)## ##int dx qquad ln(x^2x+2)## we use IBP ##int u v' = uv  int u' v## and the trick here is ##u = ln(x^2x+2),...
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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...
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Waiting for answer Find the volume of the largest rectangular box in the ﬁrst...
By solving for ##z##, ##x+2y+3z=6 Leftrightarrow z= 21/3x2/3y## 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(21/3x2/3y)=2xy1/3x^2y2/3xy^2## Second Partial Test Let us find critical points. ##V_x=2...
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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...
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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...
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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)(xa)## 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)(x1)=3+3(x1)=3x6##. 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(...
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Waiting for answer What is the line of symmetry of the graph of ##y=1/(x1)##?
The graph is a hyperbola, so there are two lines of symmetry: ##y=x1## and ##y=x+1## The graph of ##y = 1/(x1)## is a hyperbola. Hyperbolas have two lines of symmetry. both li...
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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 xvalue that makes both the denominator and the numerator equal zero. For example, the grap...
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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...
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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)##...
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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 sumtoproduct : ##costhetacosphi=2sin((theta+phi)/2)sin((thetaphi)/2)##, ##=lim_(hrarr...
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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]##
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Waiting for answer How do you integrate ##int sin^2(x) dx## using integration...
##intsin^2(x)dx=x/2sin(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)...
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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...
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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...
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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))= (x1) / (2x^(3/2))## Wolfram Computational Engine If you mean: ##sq...
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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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