AP Calculus AB Continuity and the Intermediate Value Theorem Test with Point Values (PDF)

A function has \(f(3)=5\), \(\lim_{x\to 3^-} f(x)=5\) and \(\lim_{x\to 3^+} f(x)=5\). Is \(f\) continuous at \(x=3\)? Justify with the three conditions. A second function \(g\) has \(\lim_{x\to 3} g(x)=5\) but \(g(3)\) is undefined. Is \(g\) continuous at \(3\)?

Classify the discontinuity of \(g\) at \(x=2\) in the graph, and then classify \(h(x)=\frac{x^{2}-25}{x-5}\) at \(x=5\).

Let \(f(x)=\frac{x^{2}-x-12}{x-4}\) for \(x\neq 4\) and \(f(4)=k\). Find \(k\) so that \(f\) is continuous at \(x=4\).

Let \(f(x)=3\) for \(x\le -1\), \(f(x)=ax+b\) for \(-1 < x < 2\) and \(f(x)=9\) for \(x\ge 2\). Find \(a\) and \(b\) so that \(f\) is continuous for all \(x\).

Let \(f(x)=x^{3}-3x-1\). Use the Intermediate Value Theorem to show that \(f\) has a zero in \((1,2)\). Write a complete justification.

A function \(f\) is continuous on \([0,6]\) with \(f(0)=-4\) and \(f(6)=8\). (a) Is there a \(c\) with \(f(c)=5\)? (1 point) (b) Is there necessarily a \(c\) with \(f(c)=10\)? (1 point)

The graph shows the fee \(F(t)\) in dollars for \(t\) hours of parking. (a) Find \(\lim_{t\to 1^-}F(t)\) and \(\lim_{t\to 1^+}F(t)\). (1 point) (b) Is \(F\) continuous at \(t=1\)? Which type of discontinuity? (1 point) (c) Find all values of \(t\) in \((0,3)\) where \(F\) is discontinuous. (1 point)

The temperature \(T(t)\) in a greenhouse is continuous. At 6:00 a.m. it is \(52^{\circ}\text{F}\) and at noon it is \(81^{\circ}\text{F}\). (a) Must the temperature have been exactly \(70^{\circ}\text{F}\) at some time? Justify. (2 points) (b) Can you conclude it reached \(90^{\circ}\text{F}\)? (1 point)