# Practice Problems for the Final Exam

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.
1. Find the vector and scalar projection of $$\langle 4,8 \rangle$$ onto $$\langle 2,1 \rangle$$.

$$\left\langle \dfrac{32}{5},\dfrac{16}{5}\right\rangle$$

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2. Let $$x=t+1$$, $$y=t^2-4$$. Find the Cartesian equation.

$$y=(x-1)^2-4$$

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3. Let $$x=2\sin \theta$$, $$y=3\cos\theta$$. Find the Cartesian equation.

$$\left(\dfrac{x}{2}\right)^2+\left(\dfrac{y}{3}\right)^2=1$$

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4. Find the vector equation of the line passing through the points $$(1,2)$$ and $$(-1,4)$$.

$$\mathbf{r}(t)=\langle 1-2t,2+2t\rangle$$

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5. Compute $$\tan \left( \arcsin \dfrac{2}{3} \right)$$.

$$\dfrac{2}{\sqrt{5}}$$

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6. Find the limit.
1. $$\displaystyle \lim_{x\rightarrow -2^-} \dfrac{x-1}{x^2(x+2)}$$

$$\dfrac{1}{16}$$
Video Errata: In the video, the problem shows the limit as x approaches $$+2$$, but the presenter will also solve the limit with $$-2$$.

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2. $$\displaystyle \lim_{x\rightarrow -3} \frac{x^2-x-12}{x+3}$$

$$-7$$

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3. $$\displaystyle \lim_{h\rightarrow 0} \frac{(4+h)^{-1}-4^{-1}}{h}$$

$$-\dfrac{1}{16}$$

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7. Let $$g(x)=\left\{ \begin{array}{ll}x^2-c^2 & \textrm{if } x<4\\ cx+20 & \textrm{if } x\geq 4\end{array}\right.$$. What value(s) of $$c$$ make(s) $$g(x)$$ continuous everywhere?

$$c=-2$$

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8. Find the limit.
1. $$\displaystyle \lim_{x\rightarrow \infty}\dfrac{7x^3+4x}{2x^3-x^2+3}$$

$$\dfrac{7}{2}$$

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2. $$\displaystyle \lim_{x\rightarrow -\infty} \dfrac{\sqrt{x^2+4x}}{4x+1}$$

$$-\dfrac{1}{4}$$

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3. $$\displaystyle \lim_{x\rightarrow \infty} \dfrac{5e^{3x}-e^{-3x}}{e^{3x}+3e^{-3x}}$$

$$5$$

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4. $$\displaystyle \lim_{x\rightarrow -\infty} \dfrac{5e^{3x}-e^{-3x}}{e^{3x}+3e^{-3x}}$$

$$-\dfrac{1}{3}$$

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5. $$\displaystyle \lim_{x\rightarrow \infty} \left( \ln \left(3x^2\right)-\ln \left(6x^4-3x+1\right)\right)$$

$$-\infty$$

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9. Find the derivative.
1. $$f(x)=x^3\left(4x^5+5\right)^8$$

$$f'(x)=3x^2\left(4x^5+5\right)^8+160x^7\left(4x^5+5\right)^7$$

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2. $$h(x)=\sqrt{\cos\left( \sin^2 x\right)}$$

$$h'(x)=\frac{1}{2}\left(\cos \left(\left(\sin x\right)^2\right)\right)^{-\frac{1}{2}} \left(-\sin \left(\left(\sin x\right)^2\right)\right) \cdot 2(\sin x)\cos x$$

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3. $$f(x)=\ln \left(xe^{-x}\right)$$

$$f'(x)=\dfrac{1}{x}-1$$

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4. $$g(x)=\dfrac{\ln x}{1-x}$$

$$g'(x)=\dfrac{\frac{1}{x}(1-x)+\ln x}{(1-x)^2}$$

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10. Find $$\dfrac{dy}{dx}$$ for the following equations.
1. $$e^{y}\sin x = x+xy$$

$$\dfrac{dy}{dx}=\dfrac{1+y-e^y\cos x}{e^y\sin x- x}$$

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2. $$x\sin y+\cos(2y)=\cos y$$.

$$\dfrac{dy}{dx}=\dfrac{-\sin y}{x\cos y-2\sin(2y)+\sin y}$$

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11. Differentiate $$y=\dfrac{e^x\left(x^2+2\right)^3}{\left(x+1\right)^4\left(x^2+3\right)^2}$$.

$$\dfrac{dy}{dx}=\dfrac{e^x\left(x^2+2\right)^3}{\left(x+1\right)^4\left(x^2+3\right)^2} \left( 1+ \dfrac{6x}{x^2+2}-\dfrac{4}{x+1}-\dfrac{4x}{x^2+3}\right)$$

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12. If $$x = 1−t^3$$ and $$y = t^2 −3t+1$$, find an equation of the tangent line corresponding to $$t = 2$$.

$$y=-\dfrac{1}{12}x-\dfrac{19}{12}$$

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13. If $$x = t^3 −3t^2$$ and $$y = t^3 −3t$$, find all points on the curve where the tangent line is vertical or horizontal.

Vertical: $$(0,0)$$, $$(-4,2)$$
Horizontal: $$(-2,-2)$$, $$(-4,2)$$

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14. A bacteria culture starts with 1000 bacteria, and after half an hour the population is tripled.
1. Find an expression for the number of bacteria after t hours.
2. Find the number of bacteria after 20 minutes.​​

1. $$y(t)=1000e^{(2\ln 3) t}$$
2. $$y\left(\frac{1}{3}\right)=1000e^{\frac{2}{3}\ln 3}$$

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15. Two sides of triangle have length $$5$$ m and $$4$$ m. The angle between them is increasing at a rate of $$\pi$$ rad/s. Find the rate at which the area of the triangle is increasing when the angle between the sides of fixed length is $$\dfrac{\pi}{6}$$.​​

$$\dfrac{\sqrt{3}}{3}\pi$$

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16. Use a linear approximation to find an approximate value of $$(1.99)^6$$.

$$62.08$$

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17. Given $$f(x) = 4 − x^2$$, show $$f(x)$$ satisfies the mean value theorem on the interval $$[1,2]$$ and find all $$c$$ that satisfy the conclusion of the Mean Value Theorem.

$$\dfrac{3}{2}$$

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18. The graph of the derivative $$f'$$ of a function $$f$$ is shown below.
1. On what intervals is $$f$$ increasing or decreasing?
2. At what values of $$x$$ does $$f$$ have a local maximum or minimum?
3. On what intervals is $$f$$ concave upward or downward?
4. State the $$x$$-coordinates of the points of inflection.

Increasing: $$(1,6)$$,$$(8,\infty)$$
Decreasing: $$(−\infty,1)$$,$$(6,8)$$
Local Minimum at $$x = 1, 8$$
Local Maximum at $$x = 6$$
Concave Upward: $$(−\infty, 2)$$, $$(3, 5)$$, $$(7, \infty)$$
Concave Downward: $$(2, 3)$$, $$(5, 7)$$
Inflection Points at $$x = 2,3,5,7$$

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19. Find the limit.
1. $$\displaystyle \lim_{x\rightarrow 0} \dfrac{\sin x - x}{x^3}$$​

$$-\dfrac{1}{6}$$

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2. $$\displaystyle \lim_{x\rightarrow 0^+} x\ln x$$​

$$0$$

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3. $$\displaystyle \lim_{x\rightarrow 1} \left( \frac{1}{\ln x} - \frac{1}{x-1}\right)$$​

$$\dfrac{1}{2}$$

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4. $$\displaystyle \lim_{x\rightarrow \infty} x^{3/x}$$

$$1$$

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20. Approximate the area under the curve $$f (x) = x^2 + 1$$ on the interval $$[1, 7]$$ using $$3$$ equal width-rectangles and using left endpoints.

$$76$$

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21. If $$g(x)=\displaystyle \int_0^x \! f(t)\, dt$$, where the graph of $$f(t)$$ is given below, where $$0\leq x\leq 10$$, evaluate $$g(0)$$, $$g(6)$$ and $$g(10)$$. Where is $$g(x)$$ increasing? What is the maximum value of $$g(x)$$?

$$g(0)=0$$
$$g(3)=9$$
$$g(6)=6$$
$$g(10)=12$$
Increasing: $$(0,3)$$, $$(6,10)$$
Maximum Value is 12 at $$t=10$$.

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22. Find the derivative of $$g(x)=\displaystyle \int_3^{\sin x} \frac{\cos t}{t} \, dt$$.

$$g'(x)=\dfrac{\cos(\sin(x))}{\sin x} \cos x$$

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23. Find the general indefinite integral.
1. $$\displaystyle \int \! \dfrac{1+\sqrt{x}+x}{x} \, dx$$

$$\ln |x|+2x^{\frac{1}{2}} + x + C$$

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2. $$\displaystyle \int \! \left(2x^3-6x+\dfrac{3}{x^2+1}\right) \, dx$$

$$\frac{1}{2}x^4-3x^2+3\arctan x + C$$

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24. Evaluate
1. $$\displaystyle \int_{-1}^0 \left(5x^2-4x+3\right)\, dx$$

$$\dfrac{20}{3}$$

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2. $$\displaystyle \int_{1}^2 \! \dfrac{x^6-x^2}{x^7}\, dx$$

$$\ln(2)-\dfrac{15}{64}$$

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3. $$\displaystyle \int_{0}^\pi \! \left( 5e^x+3\sin x\right)\, dx$$

$$5e^\pi +1$$

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4. $$\displaystyle \int \! x^3\cos \left(x^4+2\right) \, dx$$

$$\dfrac{1}{4}\sin\left(x^4+2\right)+C$$

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5. $$\displaystyle \int_1^e \! \dfrac{\ln x}{x}\, dx$$

$$\dfrac{1}{2}$$

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