AP Calculus AB Continuity and the Intermediate Value Theorem Test with Answer Key (Free PDF)
This continuity test for AP Calculus AB covers continuity and the intermediate value theorem (Common Core AP CALC AB 1.10, AP CALC AB 1.11, AP CALC AB 1.12, AP CALC AB 1.16), with a free PDF to print.
Key takeaway: This test helps AP Calculus AB students check continuity, discontinuity types and the Intermediate Value Theorem in 45 minutes.
- Topic: Continuity and the Intermediate Value Theorem (AP Calculus AB)
- Content: 8 questions with point values, about 45 minutes
- Standards: AP CALC AB 1.10, AP CALC AB 1.11, AP CALC AB 1.12, AP CALC AB 1.16
- Answers: On a separate page, so you can try every worksheet problem first
- Format: Free printable PDF, US Letter, black and white
This test helps AP Calculus AB students check continuity, discontinuity types and the Intermediate Value Theorem in 45 minutes. Also, it has 8 questions worth 20 points, and every question rewards a clear justification.
Also, the questions cover the three conditions, classifying discontinuities, repairing a function, continuity at two points, a root guaranteed by the theorem, which values are guaranteed, a parking fee with steps, and a temperature model. Write each condition by name and state yes or no, since AP graders look for the exact reasoning chain.
This test checks continuity, types of discontinuity, making piecewise functions continuous and the Intermediate Value Theorem (AP Calculus AB, 1.10, 1.11, 1.12, 1.16). It has 8 questions worth 20 points in total. Allow 45 minutes, work without notes, and show all work, since justification earns points on the AP exam.
Time and points
- Grade: AP Calculus AB
- Chapter: Continuity and the Intermediate Value Theorem
- Standards: AP CALC AB 1.10, AP CALC AB 1.11, AP CALC AB 1.12, AP CALC AB 1.16
- Time: 45 minutes
- Total: 20 points
Continuity test: questions
1 – Three conditions [2 pts]
First, 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 \(3\)?
2 – Classifying discontinuities [2 pts]
Classify the discontinuity of \(g\) at \(x=2\) in the graph, and then classify \(h(x)=\frac{x^{2}-25}{x-5}\) at \(x=5\).
3 – Repairing a function [2 pts]
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 \(x=4\).
4 – Continuity at two points [3 pts]
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 \(x\).
5 – IVT and a root [3 pts]
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.
6 – IVT guarantees [2 pts]
First, 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)
Question 7 and after
7 – Parking fee steps [3 pts]
Next, 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)
8 – Temperature and the IVT [3 pts]
Then, 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)
Key facts behind this continuity test
- Continuity needs three things: f(a) exists, the limit exists, and they are equal.
- Removable discontinuities are holes; jump discontinuities have different one-sided limits; infinite discontinuities are vertical asymptotes.
- To make a piecewise function continuous, set the one-sided limits equal at the boundary.
- Then, the IVT guarantees a value, but only if the function is continuous on the whole closed interval.
Common mistakes with this continuity test
- Students solve one boundary and assume the other follows. Each boundary needs its own equation, and the two equations together determine both constants.
- Students find a root by trial and skip the theorem. The question wants a justification, so state continuity, the sign change, and the conclusion.
- Students call the parking fee continuous because it is defined everywhere. A step function jumps at each change in price, so it has jump discontinuities.
Test-taking strategy
- Begin with questions 1, 2, 3 and 6, which are worth 2 points and take little time. Save the 3-point questions for when the quick points are secured.
- For the theorem, write a fixed three-line template: function is continuous, value at one end is negative and at the other positive, so a zero exists. Then, use it every time.
- On the last pass, check every discontinuity classification against its limits. Also, a mislabeled jump is a common error that a quick look can catch.
Download the continuity test as a PDF with the button below and keep it with your notes.
Continuity test: frequently asked questions
What do I write to apply the Intermediate Value Theorem?
State that the function is continuous on the closed interval, give the values at both endpoints, and name the target value between them. Then conclude that some c in the interval gives that value. All three parts are needed.
Does the theorem tell me where the root is?
No. Also, it guarantees that a root exists in the interval but does not locate it. To estimate the location, narrow the interval by testing midpoints. Also, the test question asks only for existence.
How do I handle a temperature model on the test?
Check that the function is continuous on the time interval, then evaluate it at both endpoints. If the target temperature is between those values, the theorem guarantees a time when it occurs. Then, include units in the conclusion.
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AP Calculus AB Continuity and the Intermediate Value Theorem Test with Answer Key
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