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Engineering Mathematics I
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Practice Problems for Exam 2
Practice Problems for Exam 2
Directions.
The following are review problems. We recommend you work the problems yourself, and then click "Answer" to check your answer. If you do not understand a problem, you can click "Video" to learn how to solve it. You can also follow the link for the full video page to find related videos.
Find the derivative.
\(f(x)=e^{x^2}\)
\(f(x)=x\sin^7(\cos(6x))\)
Answer
\(f'(x)=e^{x^2}\cdot 2x\)
\(f'(x)=\sin^7(\cos(6x))+x\cdot7\sin^6(\cos(6x))\cdot\cos(\cos(6x))\cdot(-\sin(6x))\cdot6\)
Video
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MLC WIR 20B M151 week6 #1ab
\(f(x)=\dfrac{(x-1)^2}{e^{x^2+2x}}\)
\(f(x)=x\sec^4(5x)\)
Answer
\(f'(x)=\dfrac{2(x-1)\cdot e^{x^2+2x}-(x-1)^2\cdot e^{x^2+2x}\cdot(2x+2)}{e^{2x^2+4x}}\)
\(f'(x)=\sec^4(5x)+x\cdot 4\sec^3(5x)\cdot\sec(5x)\cdot\tan(5x)\cdot5\)
Video
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MLC WIR 20B M151 week6 #1cd
\(f(x)=\cos(x+e^{3x})\)
\(f(x)=\dfrac{(4-x)^2}{\tan x}\)
Answer
\(f'(x)=-\sin\left(x+e^{3x}\right)\cdot\left(1+3e^{3x}\right)\)
\(f'(x)=\dfrac{2(4-x)\cdot(-1)\cdot\tan x-(4-x)^2\cdot \sec^2 x}{\tan^2x}\)
Video
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MLC WIR 20B M151 week6 #1ef
\(f(x)=\ln \left(\sin^2x\right)\)
\(g(x)=\ln \left(xe^{-2x}\right)\)
Answer
\(f'(x)=\dfrac{2\sin x\cdot \cos x}{\sin^2 x} = \dfrac{2\cos x}{\sin x}\)
\(g'(x)=\dfrac{e^{-2x}+x\cdot e^{-2x}(-2)}{x\cdot e^{-2x}}\)
Video
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MLC WIR 20B M151 week6 #1gh
\(f(x)=\log_5(1+\cos x)\)
\(f(x)=\arcsin\left(\dfrac{1}{x}\right)\)
Answer
\(f'(x)=\dfrac{-\sin x}{(1+\cos x)\ln 5}\)
\(f'(x)=\dfrac{1}{\sqrt{1-\left(\dfrac{1}{x}\right)^2}}\cdot \dfrac{-1}{x^2}\)
Video
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MLC WIR 20B M151 week6 #1ij
\(f(x)=\sqrt{1-x^2}\arcsin x\)
\(f(x)=\arctan \left(x^2-x\right)\)
Answer
\(f'(x)=\dfrac{1}{2}\left(1-x^2\right)^{-1/2} \cdot (-2x)\cdot \arcsin x + 1\)
\(f'(x)=\dfrac{1}{1+\left(x^2-x\right)^2} \cdot (2x-1)\)
Video
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MLC WIR 20B M151 week6 #1kl
Find \(\dfrac{dy}{dx}\).
\(x^2y^3-5x^3=\sec(4y)+10^{y^2}\)
Answer
\(\dfrac{dy}{dx}=\dfrac{2xy^3-15x^2}{4\sec(4y)\tan(4y)+2y\cdot 10^{y^2}\ln 10-3x^2y^2}\)
Video
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MLC WIR 20B M151 week6 #2a
\(\tan\left(xy^2\right) +\sin y = 6x^2+8y+2\)
Answer
\(\dfrac{dy}{dx}=\dfrac{12x-y^2\cdot\sec^2\left(xy^2\right)}{2xy\cdot \sec^2\left(xy^2\right)+\cos y-8}\)
Video
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MLC WIR 20B M151 week6 #2b
If \(f(x)=\cos \left( \ln x^2\right)\), find \(f'(1)\).
Answer
\(f'(1)=0\)
Video
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MLC WIR 20B M151 week6 #3
Use logarithmic differentiation to find the derivative of the function.
\(y=x^{\cos x}\).
Answer
\(\dfrac{dy}{dx}=x^{\cos x} \left(-\sin x \ln x + \dfrac{\cos x}{x}\right)\)
Video
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MLC WIR 20B M151 week6 #4a
\(y=(\ln x)^{\cos \left(x^2+3\right)}\)
Answer
\(y'=(\ln x)^{\cos \left(x^2+3\right)} \left(-2x\sin \left(x^2+3\right) \cdot \ln (\ln x) + \dfrac{\cos \left(x^2+3\right)}{x\ln x}\right) \)
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MLC WIR 20B M151 week6 #4b
Given that \(h(5)=3\), \(h'(5)=-2\), \(g(5)=-3\) and \(g'(5)=6\), find \(f'(5)\) for each of the following functions.
\(f(x)=g(x)h(x)\).
Answer
\(f'(5)=24\)
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MLC WIR 20B M151 week6 #5a
\(f(x)=\dfrac{g(x)}{h(x)}\)
Answer
\(f'(5)=\dfrac{4}{3}\)
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MLC WIR 20B M151 week6 #5b
\(f(x)=g(h(x))\)
Answer
Not enough information
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MLC WIR 20B M151 week6 #5c
Find \(h''(1)\) if \(h(x)=e^{-x^2}\).
Answer
\(\dfrac{2}{e}\)
Video
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MLC WIR 20B M151 week6 #6
The vector function \(\mathbf{r}(t)=(t+e^t)\mathbf{i}-2\sin(t)\mathbf{j}\) represents the position of a particle at time \(t\). Find the speed of the object at the point \((1,0)\).
Answer
The speed is \(2\sqrt{2}\)
Video
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MLC WIR 20B M151 week6 #7
At what point on the graph of \(f(x)=\sqrt{x}\) is the tangent line parallel to the line \(2x-3y=4\)?
Answer
\(\left(\dfrac{9}{16},\dfrac{3}{4}\right)\)
Video
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MLC WIR 20B M151 week6 #8
For what values of \(x\), with \(0\leq x\leq 2\pi\), does the curve \(y=x+\dfrac{1}{3}\cos(3x)\) have a horizontal tangent?
Answer
\(x=\dfrac{\pi}{6},\dfrac{5\pi}{6}, \dfrac{3\pi}{2}\)
Video
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MLC WIR 20B M151 week6 #9
Find the \(77\)th derivative of \(g(x)=\sin(2x)\).
Answer
\(g^{(77)}(x)=2^{77}\cos(2x)\)
Video
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MLC WIR 20B M151 week6 #10
Use a linear approximation at \(x=1\) to approximate the value of \(\sqrt{1.1}\).
Answer
\(\dfrac{21}{20}\)
Video
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MLC WIR 20B M151 week6 #11
A particle moves according to the equation \(s(t)=t^2-t\), where \(t\) is measured in seconds and \(s\) in feet. What is the total distance the particle travels during the first \(2\) seconds?
Answer
The total distance is \(2.5\) ft.
Video
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MLC WIR 20B M151 week6 #12
Find the point(s) on the curve \(x=t^2+4t\), \(y=t^2+2t\) where the tangent line is vertical or horizontal.
Answer
Horizontal at \((-3,1)\)
Vertical at \((-4,0)\)
Video
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MLC WIR 20B M151 week6 #13
Find the equation of the tangent line to the curve parametrized by \(x=5t-t^3\), \(y=t^2-2t\) at the point corresponding to \(t=0\).
Answer
\(y=-\dfrac{2}{5}x\)
Video
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MLC WIR 20B M151 week6 #14
Find the equation of the tangent line to the curve \(2x^2y-3y^2=-11\) at the point \((2,-1)\).
Answer
\(y=\dfrac{4}{7}x-\dfrac{15}{7}\)
Video
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MLC WIR 20B M151 week6 #15
For the following function, find the value of \(a\) and \(b\) that make \(f(x)\) differentiable everywhere. \[f(x)=\left\{\begin{array}{ll}ax^2+x+1 & \textrm{if } x\leq -1\\ bx-1 & \textrm{if } x> -1\end{array} \right.\]
Answer
\(a=2, \quad b=-3\)
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MLC WIR 20B M151 week6 #16
Suppose the linear approximation for a function \(f(x)\) at \(a=2\) is given by the tangent line \(y=-3x+11\). If \(g(x)=(f(x))^2\), find the linear approximation for \(g(x)\) at \(a=2\).
Answer
\(L(x)=25-30(x-2)\)
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MLC WIR 20B M151 week6 #17