Mathematics 265 Introduction to Calculus I

Sample Final Exam

Sample Final Examination 2

Time: 3 hours
Passing grade: 55%
Total points: 81

1. (10 points)

1. Evaluate the following limits, no credit will be given for unjustified answers. If a limit does not exist explain why.

   1. $\underset{x\to \infty }{\lim }\left({\displaystyle \frac{3x^2-4}{4+2x+2x^4}}\right)\sin \left(x\right)$

   2. $\underset{x\to 5\pi ∕2}{\lim }\tan x$

2. (8 points)

2. Give the indicated derivatives listed below. State which rule(s) you are applying. You may not need to simplify your answer.

   1. ${\displaystyle \frac{d}{dx}}{\displaystyle \frac{\tan (2x)}{\sqrt{x}}}$
   2. ${{\displaystyle \frac{d}{dx}}{\cos }^3(x^2)|}_{x=\sqrt{\pi }∕2}$
3. (4 points)

3. 1. Use the Extreme Value Theorem to find the absolute extreme values of the function

      $f(x)={\displaystyle \frac{1}{x^2+1}}$   on the interval $[-1,1]$.

   2. Use the properties of the definite integral to estimate the value of the integral

      ${\displaystyle {\int }_{-1}^1\frac{1}{x^2+1}dx}$.

4. (8 points)

4. Give the interpretation in terms of the graph of the function $f\left(x\right)$ if

   1. $f\left(0\right)=-3$

   2. $\underset{x\to \infty }{\lim }f\left(x\right)=-2$

   3. $\underset{x\to 3}{\lim }f\left(x\right)=\infty$

   4. $f^{\prime }(x)<0$ on the interval $\left[3,\infty \right)$

   5. $f^{\prime }\left(x\right)<0$ on the interval $\left(-\infty ,-2\right)$

   6. $f^{″}\left(x\right)>0$ on the interval $\left[4,\infty \right)$

   Give a sketch of the graph of $f\left(x\right)$.

5. (10 points)

5. What are the dimensions of the cheapest rectangular box that can be constructed if the material for the box is $1.20 per square cm and the cost for the lid is $1.50 per square cm. The length of the base is twice as long as it is wide and the volume must be 120 cm²? What is the minimum cost?

6. (10 points)

6. Integrate each one of the integrals listed below. Indicate the technique you are using.

   1. ${\displaystyle \int \frac{\cos (\sqrt{2x})}{\sqrt{x}}}dx$

   2. ${\displaystyle {\int }_0^{\pi /3}\tan x{\sec }^{2}xdx}$

   3. ${\displaystyle \int \frac{x^3+\sqrt{5x}-4}{x^2}}dx$

   4. ${\displaystyle \int {\sec }^{3}x\tan xdx}$

      Hint: ${\sec }^{3}x\tan x={\sec }^{2}x\sec x\tan x$.

7. (6 points)

7. 1. Find the absolute extreme values of the function $f\left(x\right)={\left(x^2+2x\right)}^{2∕3}$ on the interval $\left[-2,3\right]$.

   2. Use the properties of definite integrals to bound de value of the integral

      ${\displaystyle {\int }_{-2}^3{(x^2+2x)}^{2/3}dx}$

8. (6 points)
8. Find the area between the curves $\cos x$ and the horizontal line $y={\displaystyle \frac{1}{\sqrt{2}}}$ in the interval $\left[-{\displaystyle \frac{\pi }{2}},{\displaystyle \frac{\pi }{2}}\right]$.
9. (4 points)
9. Assume that 20 ft-lb of work is required to stretch a spring 1 ft beyond its natural length.
   1. What is the spring constant?
   2. How much work is required to stretch the spring 2 ft beyond its natural length?
10. (6 points)
10. A particle moves along a line so that its velocity at time $t$ is $v\left(t\right)=t^2-3t+2$ (meter/sec).
    1. Find the displacement of the particle during the time period $\left[0,3\right]$.
    2. Find the distance traveled during this same time period.
11. (5 points)
11. A sprinter in a 100 m race explodes out of the starting block with an acceleration of 4 m/s², which she sustains for 2 seconds. Her acceleration then drops to zero for the rest of the race. What is her time for the race?
12. (4 points)
12. Find a positive number $k$ such that the average value of the function $f\left(x\right)={\displaystyle \frac{5}{x^2}}$ over the interval between ${\displaystyle 1}$ and ${\displaystyle k}$ is 32.