Mathematics 265 Introduction to Calculus I

Sample Midterm Exam

Sample Midterm Examination 2

Time: 3 hours
Passing grade: 55%
Total points: 94

1. (4 points)
1. Determine which of the following equations are well defined functions with an independent variable $x$. Explain.
   1. ${\displaystyle \frac{2y}{x^2-3\left|x\right|}}=1$
   2. $y^2+x^2=10$
2. (4 points)
2. Consider a cone whose height is double the size of its radius.
   1. Express the volume $V$ of the cone as a function of radius $r$.
   2. Find the domain of the function in a.
   3. Use the function in a. to compute the volume of the cone for $r=3$.

   Hint: The volume of a cone is $V={\displaystyle \frac{\pi r^{2}h}{3}}$.

3. (4 points)
3. The population $P$ of a city in the year $y$ is $P\left(y\right)$. Write a sentence (in layman’s terms) that explains the meaning of the expressions
   1. ${\displaystyle \frac{p\left(15\right)-P\left(2\right)}{13}}=1\text{,}431$
   2. $P^{\prime }\left(10\right)=14\text{,}000$.
4. (6 points)
4. Indicate the transformations in the order they are applied to the basic graph of $\left|x\right|$ in the interval in order to obtain the graph of the function $f\left(x\right)=-3|x-2|$. Apply the transformations to sketch the graph of $f$, no credit will be given if other method is used.
5. (10 points)
5. Interpret in terms of the graph of the function $f$ the conditions listed below. Sketch the graph of a single function $f\left(x\right)$ which satisfies all of the conditions listed below.
   1. $\underset{x\to \infty }{\lim }f(x)=3$
   2. $$ f(0)=0$$
   3. $\underset{x\to -1}{\lim }f(x)=2$
   4. $\underset{x\to 1}{\lim }f\left(x\right)=\infty$
   5. the function is continuous but not differentiable at $x=0$
6. (10 points)

6. Find the vertical and horizontal asymptotes of the function

   $f\left(x\right)={\displaystyle \frac{x^2-3x-4}{x^2-16}}.$

7. (24 points)

7. Evaluate each of the limits given below. If a limit does not exist, explain why.

   Note: No credit will be given for unjustified answers.

   1. $\underset{x\to 1}{lim}{\displaystyle \frac{x^2-1}{x^3-1}}$
   2. $\underset{x\to 0}{lim}{\displaystyle \frac{x^2}{1-\cos \left(2x\right)}}$
   3. $\underset{x\to 0}{lim}{\displaystyle \frac{6x-9}{x^3-12x+3}}$
   4. $\underset{x\to \infty }{lim}{\displaystyle \frac{5-2x^3}{x^2+2}}$
   5. $\underset{x\to -\pi ∕3}{lim}{\displaystyle \frac{\tan \left(2x\right)}{3x+\pi }}$
   6. $\underset{x\to 2}{lim}{\displaystyle \frac{\cos \left(\pi x\right)}{{\left(x-2\right)}^2}}$
8. (6 points)
8. Use linearization to estimate the value of $\sin \left(62^o\right)$.
9. (16 points)

9. Find each of the derivatives listed below. Show your work.

   Note: You may not need to simplify your answer.

   1. ${\displaystyle \frac{d^2}{d{x}^2}}\cot \left(2x\right)$

   2. ${\displaystyle \frac{d}{dx}}\sec \left(x^2-3x\right)$

   3. ${\displaystyle \frac{d}{dx}}{\displaystyle \frac{x^2-\cos \left(3x\right)}{x\sin \left(2x\right)}}$

   4. ${\displaystyle \frac{d}{dx}}{{\displaystyle \frac{f(x)}{g(x)}}|}_{x=1}$ where $f\left(1\right)=3$, $g\left(1\right)=6$, $f^{\prime }\left(x\right)=\sqrt{x}$ and $g^{\prime }\left(x\right)=3\sin \left(\pi x/3\right)$.
10. (4 points)
10. Find the equation of the tangent line to the curve $y^3+y{x}^2+x^2=3y^2$ at the point (1,1).
11. (6 points)
11. A rocket, rising vertically, is tracked by a radar station that is on the ground 5 mi from the launch pad. How fast is the rocket rising when it is 4 mi high and its distance from the radar station is increasing at a rate of 2000 mi/h?