Mathematics 265 Introduction to Calculus I

Sample Midterm Exam

Sample Midterm Examination 1

Time: 3 hours
Passing grade: 55%
Total points: 64

1. Give the exact value of ${\displaystyle cos \left(-\frac{\pi }{12}\right)\text{.}}$ (5 points)
2. Let ${\displaystyle f\left(x\right)=2x^2-5}$ and ${\displaystyle g\left(x\right)=\frac{x+5}{2x-9}\text{.}}$ (6 points)

   Find the composite functions and their corresponding domains.

   1. ${\displaystyle f\left(g\left(x\right)\right)}$
   2. ${\displaystyle g\left(f\left(x\right)\right)}$
3. Give a labeled graph of the function ${\displaystyle g\left(x\right)=3{\left(x+4\right)}^2}$ by starting with the graph of a basic function, and then applying the appropriate transformations. (5 points)

   Explain the procedure you are using.

   Note: No credit will be given if any other method is used.

4. Evaluate each of the limits below. If a limit does not exist explain why. (16 points)
   1. ${\displaystyle \underset{x\to 2}{lim}\left(3x^2-2x+1\right)}$
   2. ${\displaystyle \underset{x\to -2}{lim}\frac{3x^2-2x-16}{{\left(x+2\right)}^2}}$
   3. ${\displaystyle \underset{x\to 3}{lim}\frac{2x^2-x+1}{x-3}}$
   4. ${\displaystyle \underset{x\to 2}{lim}\frac{\sqrt{6-x}-2}{\sqrt{3-x}-1}}$
   5. ${\displaystyle \underset{x\to 0}{lim}\frac{sin \left(3x\right)}{x^2-x}}$
5. Compute the derivatives of each of the functions below. You may not need to simplify your answers. (8 points)
   1. ${\displaystyle y=4x^5+3x^4-6x^3+6}$
   2. ${\displaystyle y=\frac{2x-16}{{\left(x+3\right)}^2}}$
   3. ${\displaystyle y=sin \left(2x^2-x+1\right)}$
   4. ${\displaystyle y={\left(-4x^3-x^2+3x+7\right)}^4}$
6. Find the values of ${\displaystyle x}$ such that tangent line to the curve ${\displaystyle f\left(x\right)=3x^2+4x-3}$ is perpendicular to the line ${\displaystyle y=6x+2\text{.}}$ (4 points)
7. Gravel is being dumped from a conveyor belt at a rate of ${\displaystyle 0.5{\text{ m}}^3\text{/min}\text{.}}$ It forms a pile in shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 4 m high? (5 points)
8. Find ${\displaystyle y^{\prime }}$ using implicit differentiation: (5 points)

   ${\displaystyle x^{3}y+x{y}^2=4xy+7\text{.}}$

9. Use differentials to find the approximate value of ${\displaystyle \sqrt{9\text{.}2}\text{.}}$ (5 points)
10. Sketch the graph of a single function ${\displaystyle f\left(x\right)}$ that satisfies all of the conditions listed below. (5 points)
    • the limit ${\displaystyle \underset{x\to 0}{lim}f\left(x\right)}$ does not exist.
    • ${\displaystyle f\left(0\right)=0\text{.}}$
    • ${\displaystyle \underset{x\to \infty }{lim}f\left(x\right)=-1\text{.}}$
    • the function ${\displaystyle f}$ is not differentiable at ${\displaystyle x=-2\text{.}}$