Calculus I -- Spring 2017 -- midterm

Name:_________________________ $\qquad$ 10 problems, 100 points total

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

$\bullet$ 1. Give an example of an even function, an odd function, and a function that's neither. Provide formulas. (Just the answer)

$\bullet$ 2. The graph of $f$ is given below. It has asymptotes. Describe them as limits. (Just the answer)

$\bullet$ 3. Find the horizontal asymptote of the function: $$f(x)=\frac{x}{5x^2-x+1}.$$

$\bullet$ 4. The graph of a function $f(x)$ is given below. Estimate the values of the derivative $f'(x)$ for $x=0,4,$ and $6$. (Just the answer)

$\bullet$ 5. The graph of function $f$ is given below. (a) At what points is $f$ continuous? (b) At what points is $f$ differentiable? (Just the answer)

$\bullet$ 6. From the definition, compute the derivative of $f(x)=x^{2}+1$ at $a=2$.

$\bullet$ 7. The graph of function $f$ is given below. Sketch the graph of the derivative $f'$ of $f$. (Just the answer)

$\bullet$ 8. Calculate the derivative of $$f(x) = \frac{e^x}{e^x-1}.$$

$\bullet$ 9. Represent this function, $h(x)= \sin (x^2-1)$, as the composition of two functions. Find its derivative.

$\bullet$ 10. Use implicit differentiation to find an equation of the line tangent to the curve $x^{2}+y^{2}=4$ passing through the point $(0,-2)$.