# Practice Problems for Exam 1

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 scalar and vector projections of $$\langle -3,1\rangle$$ onto $$\langle 2,5\rangle$$. ​

Scalar Projection: $$-\dfrac{1}{\sqrt{29}}$$
Vector Projection: $$\left\langle -\dfrac{2}{29}, -\dfrac{5}{29}\right\rangle$$

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2. Vector starts at the point $$(-3,1)$$ and ends at the point $$(6,3)$$. Find a unit vector in the direction of a.

$$\left\langle \dfrac{9}{\sqrt{85}}, \dfrac{2}{\sqrt{85}}\right\rangle$$

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3. Determine a vector equation of the straight line which passes through the point $$(1,-1)$$ and is parallel to the vector $$\langle -2,3\rangle$$.

$$\left\langle 1-2t, -1+3t \right\rangle$$

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4. Consider the curve $$x(t)=t-2, y(t)=t^2-3$$.
1. Is the point $$(4,40)$$ on the graph of the curve?
2. Eliminate the parameter to find a Cartesian equation.
3. Sketch the curve.​

1. No
2. $$y=(x+2)^2-3$$
3. See Video

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5. Find a Cartesian equation for the parametric curve given by the equation $$x=\dfrac{\cos t}{2}$$, $$y=\sin t$$, $$0\leq t \leq 2\pi$$. Describe the direction of the motion as $$t$$ increases.​

$$4x^2+y^2=1$$

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6. Find the angle $$\angle ABC$$ of the triangle with the vertices, $$A(3,0)$$, $$B(0,3)$$, $$C(5,4)$$.​

$$\theta=\cos^{-1} \left( \dfrac{2}{\sqrt{13}}\right)$$

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7. Find the distance from the point $$(1,2)$$ to the line $$y=3x-4$$.

$$\dfrac{3}{\sqrt{10}}$$

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8. Two forces $$\mathbf{F}_1$$ and $$\mathbf{F}_2$$ with magnitudes 10 lb and 12 lb, respectively, act on an object at a point $$P$$ as shown in the figure. Find the resultant force $$\mathbf{F}$$ acting at $$P$$ as well as its magnitude and its direction.

Magnitude: $$\sqrt{\left(-5\sqrt{2}+6\sqrt{3}\right)^2 + \left( 5\sqrt{2}+6\right)^2}$$
Direction: $$\theta = \tan^{-1}\left(\dfrac{5\sqrt{2}+6}{-5\sqrt{2}+6\sqrt{3}}\right)$$

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9. A constant force with the vector representation $$\mathbf{F}=\langle 10, 18\rangle$$ moves an object along a straight line form the point $$(2,3)$$ to the point $$(4,9)$$. Find the work done, if the distance is measure in meters and the magnitude of the force is measured in Newtons.

128 J

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10. A woman exerts a horizontal force of 65 lb on a crate as she pushes it up a ramp that is 20 ft long and inclined at an angle of $$30^\circ$$ above the horizontal. Find the work done on the box.

$$650\sqrt{3}$$ ft$$\cdot$$lb

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11. Find the vector equation of the line perpendicular to the vector $$\langle 3,5\rangle$$ and passing through the point $$(5,1)$$.

$$\mathbf{r}(t)=\langle 5-5t, 1+3t\rangle$$

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12. Consider the line $$x=8-2t$$, $$y=14+7t$$.
1. Find a vector perpendicular to the line
2. Find the Cartesian form.
3. Sketch the graph.​

1. $$\langle-7,-2\rangle$$ or $$\langle7,2\rangle$$
2. $$y=-\dfrac{7}{2}x+42$$
3. See the video

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13. Simplify.
1. $$\tan \left(\arccos\left( \dfrac{1}{4}\right)\right)$$
2. $$\sin(\arctan(2))$$
3. $$\tan(\arcsin(3x))$$​

1. $$\sqrt{15}$$
2. $$\dfrac{2}{\sqrt{5}}$$
3. $$\dfrac{3x}{\sqrt{1-9x^2}}$$

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14. Evaluate the limit.
1. $$\displaystyle \lim_{x\rightarrow 1} \dfrac{x^2-x-2}{x+1}$$
2. $$\displaystyle \lim_{t\rightarrow 0}\dfrac{\sqrt{2-t}-\sqrt{2}}{t}$$
3. $$\displaystyle \lim_{h\rightarrow 0}\dfrac{(3+h)^{-1}-3^{-1}}{h}$$
4. $$\displaystyle \lim_{x\rightarrow -4^{-}} \dfrac{|x+4|}{x+4}$$
5. $$\displaystyle \lim_{x\rightarrow -1^{-}} \dfrac{x-2}{x+1}$$
6. $$\displaystyle \lim_{x\rightarrow -\infty} \dfrac{-x^3+2x^2-4x}{8+4x^2-5x^3}$$
7. $$\displaystyle \lim_{x\rightarrow \infty} \dfrac{e^x+2e^{-x}}{2e^x-e^{-x}}$$
8. $$\displaystyle \lim_{x\rightarrow \infty}\ln \left( \dfrac{x}{x^2-3x}\right)$$
9. $$\displaystyle \lim_{x\rightarrow \infty} \arctan\left(e^x\right)$$
10. $$\displaystyle \lim_{x\rightarrow \infty} \arctan(\ln x)$$
11. $$\displaystyle\lim_{x\rightarrow 0^+} \arctan (\ln x)$$​

1. $$-1$$
2. $$-\dfrac{1}{2\sqrt{2}}$$
3. $$-\dfrac{1}{9}$$
4. $$-1$$
5. $$\infty$$
6. $$\dfrac{1}{5}$$
7. $$\dfrac{1}{2}$$
8. $$-\infty$$
9. $$\dfrac{\pi}{2}$$
10. $$\dfrac{\pi}{2}$$
11. $$-\dfrac{\pi}{2}$$

15. Given $$-2x-2\leq f(x) \leq \dfrac{1}{2} x^2$$, compute $$\displaystyle \lim_{x\rightarrow -2} f(x)$$.​

2

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16. Find the vertical and horizontal asymptotes of $$f(x)=\dfrac{(x-1)(x+3)}{x^2-1}$$.​

Vertical Asymptote: $$x=-1$$
Horizontal Asymptote: $$y=1$$

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17. Given the graph of $$f(x)$$ below, sketch the graph of the derivative.

See the Video

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18. Given the function below, find the value of $$a$$ which makes the function continuous everywhere.$f(x)=\left\{ \begin{array}{cc} x-4a& \textrm{if } x<-2\\ax^2 & \textrm{if } x\geq -2\end{array} \right.$

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

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19. Which of the following intervals must contian a solution to the equation $$2x^3+16x+3=22$$?
1. $$[-2,-1]$$
2. $$[-1,0]$$
3. $$[0,1]$$
4. $$[1,2]$$
5. $$[2,3]$$

d. $$[1,2]$$

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20. Find the average rate of change of $$f(x)=x^2+6$$ from $$x=-3$$ to $$x=1$$.​

-2

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21. Find $$f'(x)$$ using the limit definition of the derivative for $$f(x)=3x^2-4$$.​

$$f'(x)=6x$$

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22. Find $$f'(x)$$ using the limit definition of the derivative for $$f(x)=\sqrt{3x+1}$$.​

$$f'(x)=\dfrac{3}{2\sqrt{3x+1}}$$

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23. Find $$f'(x)$$ using the limit definition of the derivative for $$f(x)=\dfrac{-2}{x+2}$$.​

$$f'(x)=\dfrac{2}{(x+2)^2}$$

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