 # Practice Problems for Exam 2

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1. Find the derivative.
1. ​$$f(x)=e^{x^2}$$
2. $$f(x)=x\sin^7(\cos(6x))$$

1. $$f'(x)=e^{x^2}\cdot 2x$$
2. $$f'(x)=\sin^7(\cos(6x))+x\cdot7\sin^6(\cos(6x))\cdot\cos(\cos(6x))\cdot(-\sin(6x))\cdot6$$

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3. ​$$f(x)=\dfrac{(x-1)^2}{e^{x^2+2x}}$$
4. $$f(x)=x\sec^4(5x)$$​

1. $$f'(x)=\dfrac{2(x-1)\cdot e^{x^2+2x}-(x-1)^2\cdot e^{x^2+2x}\cdot(2x+2)}{e^{2x^2+4x}}$$
2. $$f'(x)=\sec^4(5x)+x\cdot 4\sec^3(5x)\cdot\sec(5x)\cdot\tan(5x)\cdot5$$

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5. $$f(x)=\cos(x+e^{3x})$$
6. $$f(x)=\dfrac{(4-x)^2}{\tan x}$$​

1. $$f'(x)=-\sin\left(x+e^{3x}\right)\cdot\left(1+3e^{3x}\right)$$
2. $$f'(x)=\dfrac{2(4-x)\cdot(-1)\cdot\tan x-(4-x)^2\cdot \sec^2 x}{\tan^2x}$$

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7. $$f(x)=\ln \left(\sin^2x\right)$$
8. $$g(x)=\ln \left(xe^{-2x}\right)$$​

1. $$f'(x)=\dfrac{2\sin x\cdot \cos x}{\sin^2 x} = \dfrac{2\cos x}{\sin x}$$
2. $$g'(x)=\dfrac{e^{-2x}+x\cdot e^{-2x}(-2)}{x\cdot e^{-2x}}$$

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9. $$f(x)=\log_5(1+\cos x)$$
10. $$f(x)=\arcsin\left(\dfrac{1}{x}\right)$$​

1. $$f'(x)=\dfrac{-\sin x}{(1+\cos x)\ln 5}$$
2. $$f'(x)=\dfrac{1}{\sqrt{1-\left(\dfrac{1}{x}\right)^2}}\cdot \dfrac{-1}{x^2}$$

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11. $$f(x)=\sqrt{1-x^2}\arcsin x$$
12. $$f(x)=\arctan \left(x^2-x\right)$$​

1. $$f'(x)=\dfrac{1}{2}\left(1-x^2\right)^{-1/2} \cdot (-2x)\cdot \arcsin x + 1$$
2. $$f'(x)=\dfrac{1}{1+\left(x^2-x\right)^2} \cdot (2x-1)$$

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2. Find $$\dfrac{dy}{dx}$$.
1. $$x^2y^3-5x^3=\sec(4y)+10^{y^2}$$​

$$\dfrac{dy}{dx}=\dfrac{2xy^3-15x^2}{4\sec(4y)\tan(4y)+2y\cdot 10^{y^2}\ln 10-3x^2y^2}$$

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2. $$\tan\left(xy^2\right) +\sin y = 6x^2+8y+2$$​

$$\dfrac{dy}{dx}=\dfrac{12x-y^2\cdot\sec^2\left(xy^2\right)}{2xy\cdot \sec^2\left(xy^2\right)+\cos y-8}$$

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3. If $$f(x)=\cos \left( \ln x^2\right)$$, find $$f'(1)$$.​

$$f'(1)=0$$

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4. Use logarithmic differentiation to find the derivative of the function.
1. $$y=x^{\cos x}$$.

$$\dfrac{dy}{dx}=x^{\cos x} \left(-\sin x \ln x + \dfrac{\cos x}{x}\right)$$

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2. $$y=(\ln x)^{\cos \left(x^2+3\right)}$$

$$y'=(\ln x)^{\cos \left(x^2+3\right)} \left(-2x\sin \left(x^2+3\right) \cdot \ln (\ln x) + \dfrac{\cos \left(x^2+3\right)}{x\ln x}\right)$$

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5. Given that $$h(5)=3$$, $$h'(5)=-2$$, $$g(5)=-3$$ and $$g'(5)=6$$, find $$f'(5)$$ for each of the following functions.
1. $$f(x)=g(x)h(x)$$.

$$f'(5)=24$$

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2. $$f(x)=\dfrac{g(x)}{h(x)}$$

$$f'(5)=\dfrac{4}{3}$$

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3. $$f(x)=g(h(x))$$

Not enough information

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6. Find $$h''(1)$$ if $$h(x)=e^{-x^2}$$.

$$\dfrac{2}{e}$$

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7. 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 $$(1,0)$$.

The speed is $$2\sqrt{2}$$

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8. At what point on the graph of $$f(x)=\sqrt{x}$$ is the tangent line parallel to the line $$2x-3y=4$$?

$$\left(\dfrac{9}{16},\dfrac{3}{4}\right)$$

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9. 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?

$$x=\dfrac{\pi}{6},\dfrac{5\pi}{6}, \dfrac{3\pi}{2}$$

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10. Find the $$77$$th derivative of $$g(x)=\sin(2x)$$.

$$g^{(77)}(x)=2^{77}\cos(2x)$$

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11. Use a linear approximation at $$x=1$$ to approximate the value of $$\sqrt{1.1}$$.

$$\dfrac{21}{20}$$

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12. 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?

The total distance is $$2.5$$ ft.

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13. Find the point(s) on the curve $$x=t^2+4t$$, $$y=t^2+2t$$ where the tangent line is vertical or horizontal.

Horizontal at $$(-3,1)$$
Vertical at $$(-4,0)$$

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14. 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 $$t=0$$.

$$y=-\dfrac{2}{5}x$$

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15. Find the equation of the tangent line to the curve $$2x^2y-3y^2=-11$$ at the point $$(2,-1)$$.

$$y=\dfrac{4}{7}x-\dfrac{15}{7}$$

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16. For the following function, find the value of $$a$$ and $$b$$ that make $$f(x)$$ differentiable everywhere. $f(x)=\left\{\begin{array}{ll}ax^2+x+1 & \textrm{if } x\leq -1\\ bx-1 & \textrm{if } x> -1\end{array} \right.$

$$a=2, \quad b=-3$$

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17. 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 $$a=2$$.

$$L(x)=25-30(x-2)$$

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