AP Calculus AB 2.1 Defining Average and Instantaneous Rates of Change At a Point Questions

Juice is sold in 20-centimeter-tall bottles with horizontal cross sections parallel to the base that are circles, as shown in the

figure. The radius of the circular cross section at height h above the base of the bottle is given by a differentiable function

r, where h and r(h) are measured in centimeters. Selected values of r(h) are given in the table shown.

Let f be the continuous function defined on the closed interval [-1,3] whose graph, consisting of three line segments, is

shown. Let g be the function given by $g(x)=\int_{0}^{x} f(t) d t$.

Approximate $r^{\prime}(3.1)$ using the average rate of change of r over the interval $0 \leq h \leq 6.2$. Show the computations that

lead to your answer. Indicate units of measure.

On what intervals, if any, is g decreasing? Give a reason for your answer.

If $f(x)=\ln x$, then $\lim _{x \rightarrow 3} \frac{f(x)-f(3)}{x-3}$ is