Calculus I -- Fall 2017 -- midterm

MATH 229 -- Fall 2017 -- Midterm exam

Name:_________________________ $\qquad$ 8 problems, 10 points each

• Write the problems in the given order, each problem on a separate page.
• Show enough work to justify your answers.

$\bullet$ 1. Three straight lines are shown below. Find the equation of one of them:

$\bullet$ 2. Find the average rate of change for the function given by the following data: $$\begin{array}{r|c} x&y=f(x)\\ \hline -1&0\\ 0&2\\ 1&3\\ 2&-1\\ 3&-2\\ 4&0\\ \end{array}$$

$\bullet$ 3. The graph of a function $f$ is given below. Find the equation of a line tangent to the graph at $(0,-1)$.

$\bullet$ 4. Illustrate with plots (separately) functions with the following behavior: (a) $f(x)\to +\infty$ as $x\to 1$; (b) $f(x)\to -\infty$ as $x\to 2^+$; (c) $f(x)\to 3$ as $x\to -\infty$.

$\bullet$ 5. From the definition, compute the derivative of $f(x)=x^2+x$ at $x=-1$.

$\bullet$ 6. The graph of function $f$ is given below. Sketch the graph of the derivative $f'$ in the space under the graph of $f$. Identify all important features of the graph.

$\bullet$ 7. Find the derivative of the function $f(x)=\sqrt{x}\cdot \cos x$.

$\bullet$ 8. Find the derivative of the function $f(x)=\sin (x^3)$ at $x=1$.