Calculus I -- Fall 2017 -- final

$\bullet$ 1. Describe the behavior of the function plotted below without referring to the picture:

$\bullet$ 2. The graph of a function $f(x)$ is given below. Estimate the values of the derivative $f'(x)$ for $x=0,4,$ and $6.$

$\bullet$ 3. The graph of the derivative $f'$ of function $f$ is given below. Sketch a possible graph of the function $f$ itself in the space under the graph of $f'$. Identify all important points on the graph.

$\bullet$ 4. Find the derivative of this function: $$f(x)=\cos(x^2+\sin x).$$

$\bullet$ 5. Let $x$ represent the time passed since the car left the city. The table below tells for what values of $x$ the velocity and the acceleration of the car are positive, negative, or zero. Let $f(x)$ represent the distance of the car from the city. Sketch the graph of $f$. $$\begin{array}{c|cc} x&\text{ velocity }&\text{ acceleration }\\ \hline 0&0&+\\ 1&+&-\\ 2&0&-\\ 3&-&- \end{array}$$

$\bullet$ 6. Evaluate the Riemann sum of $f$ below on the interval $[-1,1.5]$ with $n=5$. What are its sample points? What does it estimate?

$\bullet$ 7. Evaluate: $$\int_0^1 \sqrt{3x}\, dx.$$

$\bullet$ 8. You have received the following email from your boss: "Tim, Look at the numbers in this spreadsheet. This stock seems to be inching up... Does it? If does, how fast? Thanks. -- Tom". Describe your actions.