REVIEW EXERCISES

252

In Problems 1–42, find the derivative of each function. When a, b, or n appear, they are constants.

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\(y=(ax+b)^{n}\)

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\(y=\sqrt{2ax}\)

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\(y=x\sqrt{1-x}\)

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\({y=\frac{1}{\sqrt{x^{2}+1}}}\)

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\(y=(x^{2}+4)^{3/2}\)

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\(F(x) =\frac{x^{2}}{\sqrt{x^{2}-1}}\)

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\(z={\frac{\sqrt{2ax-x^{2}}}{x}}\)

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\(y=\sqrt{x}+\sqrt[3]{x}\)

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\(y=(e^{x}-x)^{5x}\)

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\(\phi (x)={\frac{(x^{2}-a^{2})^{3/2}}{\sqrt{x+a}}}\)

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\(f(x)={\frac{x^{2}}{(x-1)^{2}}}\)

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\(u= (b^{1/2}-x^{1/2}) ^{2}\)

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\(y=x\sec (2x)\)

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\(u=\cos ^{3}x\)

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\(y=\sqrt{a^{2}\sin \left( \frac{x}{a}\right)}\)

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\(\phi (z)=\sqrt{1+\sin z}\)

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\(u=\sin v-\frac{1}{3}\sin ^{3}v\)

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\(y=\tan \sqrt{\frac{\pi }{x}}\)

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\(y= (1.05) ^{x}\)

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\(v=\ln (y^{2}+1)\)

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\(z=\ln (\sqrt{u^{2}+25} - u)\)

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\(y=x^{2}+2^{x}\)

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\(y=\ln [\sin (2x)]\)

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\(f(x)=e^{-x}\sin (2x+\pi)\)

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\(g(x)=\ln (x^{2}-2x)\)

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\(y=\ln \frac{x^{2}+1}{x^{2}-1}\)

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\(y=e^{-x}\ln x\)

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\(w=\ln \!\big(\sqrt{x+7}-\sqrt{x}\big)\)

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\(y=\frac{1}{12}\ln \left( \frac{x}{\sqrt{144-x^{2}}}\right)\)

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\(y=\ln (\tan ^{2}x)\)

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\(f(x) =\frac{e^{x}(x^{2}+4) }{(x-2)}\)

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\(y=\sin^{-1} (x-1) + \sqrt{2x-x^{2}}\)

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\(y=2\sqrt{x}-2\tan ^{-1}\sqrt{x}\)

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\(y=4\tan ^{-1}\frac{x}{2}+x\)

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\(y=\sin ^{-1}(2x-1)\)

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\(y=x^{2}\tan ^{-1}\frac{1}{x}\)

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\(y=x\tan ^{-1}x-\ln \sqrt{1+x^{2}}\)

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\(y=\sqrt{1-x^{2}}(\sin ^{-1}x)\)

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\(y=\tanh \frac{x}{2}+\frac{2x}{4+x^{2}}\)

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\(y=x\sinh x\)

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\(y=\sqrt{\sinh x}\)

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\(y=\sinh ^{-1}e^{x}\)

In Problems 43–48, find \(y'=\frac{dy}{dx}\) using implicit differentiation.

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\(x=y^{5}+y\)

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\(x=\cos ^{5}y+\cos y\)

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\(\ln x+\ln y=x\cos y\)

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\(\tan (xy)=x\)

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\(y=x+\sin (xy)\)

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\(x=\ln (\csc y+\cot y)\)

In Problems 49–52, find \(y^\prime\) and \(y^{\prime\prime}\).

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\(xy+3y^{2}=10x\)

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\(y^{3}+y=x^{2}\)

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\(xe^{y}=4x^{2}\)

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\(\ln (x+y) =8x\)

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The function \(f(x) =e^{2x}\) has an inverse function \(g.\) Find \(g^\prime (1)\).

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The function \(f(x) =\sin x\) defined on the restricted domain \(\left[ -\frac{\pi }{2},\,\frac{\pi }{2}\right]\) has an inverse function \(g.\) Find \(g^\prime \left(\frac{1}{2}\right)\).

In Problems 55–56, express each limit in terms of the number \(e\).

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\(\lim\limits_{n\rightarrow \infty }\left( 1+\frac{2}{5n}\right) ^{n}\)

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\(\lim\limits_{h\rightarrow 0} ( 1+3h) ^{2/h}\)

In Problems 57 and 58, find the exact value of each expression.

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\(\sinh 0\)

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\(\cosh ( \ln 3)\)

In Problems 59 and 60, establish each identity.

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\(\sinh x+\cosh x=e^{x}\)

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\(\tanh (x+y) =\frac{\tanh x+\tanh y}{1+\tanh x\tanh y}\)

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If \(f(x)=\sqrt{1-\sin ^{2}x}\), find the domain of \(f^\prime\).

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If \(f(x)=x^{1/2}(x-2)^{3/2}\) for all \(x ≥ 2\), find the domain of \(f^\prime\).

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Let \(f\) be the function defined by \(f(x)=\sqrt{1+6x}.\)

  1. What are the domain and the range of \(f\)?
  2. Find the slope of the tangent line to the graph of \(f\) at \(x=4\).
  3. Find the \(y\)-intercept of the tangent line to the graph of \(f\) at \(x=4\).
  4. Give the coordinates of the point on the graph of \(f\) where the tangent line is parallel to the line \(y=x+12\).

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Tangent and Normal Lines Find equations of the tangent and normal lines to the graph of \(y=x\sqrt{x+(x-1)^{2}}\) at the point \((2,\,2 \sqrt{3})\).

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Find the differential \(dy\) if \(x^{3}+2y^{2}=x^{2}y\).

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Measurement Error If \(p\) is the period of a pendulum of length \(L\), the acceleration due to gravity may be computed by the formula \( g=\frac{(4\pi ^{2}L)}{p^{2}}\). If \(L\) is measured with negligible error, but a \(2{\%}\) error may occur in the measurement of \(p\), what is the approximate percentage error in the computation of \(g\)?

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Linear Approximation Find a linear approximation to \[ \hbox{\(y=x+\ln x\) at \(x=1\).} \]

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Measurement Error If the percentage error in measuring the edge of a cube is \(5{\%}\), what is the percentage error in computing its volume?

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For the function \(f(x) =\tan x\):

  1. Find the differential \(dy\) and \(\Delta y\) when \(x=0.\)
  2. Compare \(dy\) to \(\Delta y\) when \(x=0\) and (i) \(\Delta x=0.5\), (ii) \(\Delta x=0.1\), and (iii) \(\Delta x=0.01\).

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For the function \(f(x) =\ln x\):

  1. Find the differential \(dy\) and \(\Delta y\) when \(x=1.\)
  2. Compare \(dy\) to \(\Delta y\) when \(x=1\) and (i) \(\Delta x=0.5\), (ii) \(\Delta x=0.1\), and (iii) \(\Delta x=0.01\).

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If \(f(x)=(x^{2}+1)^{(2-3x)}\), find \(f^\prime (1)\).

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Find \(\lim\limits_{x\rightarrow 2}\frac{\ln x-\ln 2}{x-2}\).

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Find \(y^{\prime }\) at \(x=\frac{\pi }{2}\) and \(y=\pi\) if \(x\sin y+y\cos x=0\).

In Problems 74–77, find the Taylor Polynomial \(P_{n}( x)\) for \(f\) at \(x_{0}\) for the given \(n\) and \(x_{0}\).

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\(f(x)=e^{2x};\quad n=4, x_{0}=3\)

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\(f(x)=\tan x;\quad n=4, x_{0}=0\)

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\(f(x)=\frac{1}{1+x}; \quad n=4, x_{0}=1\)

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\(f(x)=\ln x; n=6, x_{0}=2\)

In Problems 78 and 79, for each function:

  1. Use the Intermediate Value Theorem to confirm that a zero exists in the given interval.
  2. Use Newton’s Method to find \(c_{3}\), the third approximation to the real zero.

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\(f(x)=8x^{4}-2x^{2}+5x-1\), interval: \((0,1)\). Let \(c_{1}=0.5.\)

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\(f(x)=2-x+\sin x\), interval: \(\left(\frac{\pi }{2},\pi \right)\). Let \(c_{1}=\frac{\pi}{2}.\)

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  1. Use the Intermediate Value Theorem to confirm that the function \(f(x) =2\cos x-e^{x}\) has a zero in the interval \((0,1)\).
  2. Use graphing technology with Newton’s Method to find \(c_{5}\), the fifth approximation to the real zero. Use the midpoint of the interval for the first approximation \(c_{1}\).

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Tangent Line Find an equation of the tangent line to the graph of \(4xy-y^{2}=3\) at the point \(( 1, 3)\).