AP Calculus AB Fundamental Theorem of Calculus and Antiderivatives Test with Point Values (PDF)
This ftc test for AP Calculus AB covers fundamental theorem of calculus and antiderivatives (Common Core AP CALC AB 6.4, AP CALC AB 6.5, AP CALC AB 6.6, AP CALC AB 6.7, AP CALC AB 6.8), with a free PDF to print.
This test checks accumulation functions, both parts of the Fundamental Theorem of Calculus, basic antiderivatives and initial value problems (AP Calculus AB, 6.4 to 6.8). 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: Fundamental Theorem of Calculus and Antiderivatives
- Standards: AP CALC AB 6.4, AP CALC AB 6.5, AP CALC AB 6.6, AP CALC AB 6.7, AP CALC AB 6.8
- Time: 45 minutes
- Total: 20 points
Ftc test: questions
Question 1 – Indefinite integral [2 pts]
Find \(\int\left(8x^{3}-6x+\frac{1}{x}\right)dx\).
Question 2 – Definite integral [2 pts]
Evaluate \(\int_{0}^{\pi/2}(\cos x+2)\,dx\).
Question 3 – FTC with the chain rule [3 pts]
Let \(G(x)=\int_{1}^{x^{3}}\sqrt{1+t^{2}}\,dt\). (a) Find \(G^{\prime}(x)\). (2 points) (b) Find \(G^{\prime}(1)\). (1 point)
Question 4 – Accumulation from a graph [3 pts]
The graph shows \(f\). Let \(g(x)=\int_{0}^{x}f(t)\,dt\). (a) Find \(g(2)\) and \(g(-2)\). (2 points) (b) At what \(x\) in \((-2,4)\) does \(g\) have a relative maximum? Justify. (1 point)
Question 5 – Initial value problem [2 pts]
Find \(f(x)\) if \(f^{\prime}(x)=4x^{3}-3\) and \(f(2)=10\).
Question 6 – Motion with integrals [3 pts]
A skater has velocity \(v(t)=12-3t\) ft/s for \(0\le t\le 6\) and position \(s(0)=5\). (a) Find \(s(t)\). (1 point) (b) Find the displacement on \([0,6]\) with an integral. (1 point) (c) Find the total distance. (1 point)
Question 7 – Net change in context [3 pts]
Oil leaks from a tank at the rate \(r(t)=20e^{-0.2t}\) gallons per hour. (a) Write an integral for the amount that leaks during the first 10 hours. (1 point) (b) Evaluate it exactly and to two decimals. (2 points)
Question 8 – Properties [2 pts]
If \(\int_{2}^{6}f(x)\,dx=9\) and \(F^{\prime}=f\) with \(F(2)=4\), find \(F(6)\) and \(\int_{6}^{2}3f(x)\,dx\).
Key facts behind this ftc test
- FTC Part 1: d/dx of the integral from a to x of f(t) dt equals f(x).
- FTC Part 2: the integral from a to b of f equals F(b) – F(a) for any antiderivative F.
- For an upper limit g(x), multiply by g′(x) (chain rule).
- Every indefinite integral needs + C; an initial condition determines C.
How to use this ftc test
Time yourself: the test is designed for an average student in 45 minutes. Answer the questions you are sure of first, then come back to the longer ones and check your units.
All AP Calculus AB chapters: AP Calculus AB math.
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