AP Calculus AB The Derivative as a Limit Test with Point Values (PDF)

Let \(f(x)=x^{2}\). (a) Find the average rate of change on \([2,5]\). (1 point) (b) Find the average rate on \([2,2.01]\) and use it to estimate \(f^{\prime}(2)\). (1 point)

Use the limit definition to find \(f^{\prime}(2)\) for \(f(x)=x^{2}-3x\).

Use the limit definition to find \(f^{\prime}(3)\) for \(f(x)=\frac{2}{x}\).

For \(g(x)=x^{2}-2x\), use \(g^{\prime}(3)=4\) to write the equation of the tangent line at \(x=3\).

Let \(k(x)=\frac{x^{2}}{2}\) for \(x\le 2\) and \(k(x)=2x-2\) for \(x > 2\). (a) Is \(k\) continuous at \(x=2\)? (1 point) (b) Compare the slopes on each side and decide whether \(k\) is differentiable at \(x=2\). (2 points)

The limit \(\lim_{h\to 0}\frac{(3+h)^{4}-81}{h}\) equals \(f^{\prime}(a)\) for some function \(f\) and number \(a\). Identify \(f\) and \(a\), then evaluate the limit.

The profit in dollars for selling \(n\) pizzas in an hour is \(P(n)=-n^{2}+30n\). (a) Find the average rate of change from \(n=10\) to \(n=12\). (1 point) (b) Use the limit definition to find \(P^{\prime}(10)\). (2 points)

Using the graph, tell whether \(f^{\prime}\) is positive, negative or zero at each point. (2 points) Point A is at \(x=0\), point B near \(x=1.2\), and point C at \(x=-1\).