AP Calculus AB Fundamental Theorem of Calculus and Antiderivatives Test with Answer Key (Free PDF)

Q: How do I answer a question that asks for the value of the accumulation function at a point?

Compute the signed area from the starting point to the given x using the graph. Treat regions above the axis as positive and below as negative. Next, add the pieces, using geometry for triangles and rectangles.

Q: How do I use the properties of integrals?

You can reverse limits with a sign change, split an interval at a point, and pull constants out. Also, these facts let you combine given values. For instance, the integral from 0 to 5 equals the integral from 0 to 2 plus the one from 2 to 5.

Fundamental theorem of calculus test – Maths PDF

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Sommaire

This fundamental theorem of calculus 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.

Key takeaway: Also, this test helps AP Calculus AB students check their use of antiderivatives and the Fundamental Theorem of Calculus in 45 minutes.

Topic: Fundamental Theorem of Calculus and Antiderivatives (AP Calculus AB)
Content: 8 questions with point values, about 45 minutes
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
Answers: On a separate page, so you can try every worksheet problem first
Format: Free printable PDF, US Letter, black and white

Also, this test helps AP Calculus AB students check their use of antiderivatives and the Fundamental Theorem of Calculus in 45 minutes. Also, it has 8 questions worth 20 points, and several questions require a written explanation as well as a calculation.

Also, the questions cover an indefinite integral, a definite integral, a derivative of an integral with the chain rule, accumulation from a graph, an initial value problem, motion, net change in context and properties of integrals. Next, write the antiderivative before substituting, and label every answer with units where the context gives them.

Then, 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

About this test

Next, Grade: AP Calculus AB
In addition, Chapter: Fundamental Theorem of Calculus and Antiderivatives
Now, 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
So, Time: 45 minutes
Also, Total: 20 points

Fundamental theorem of calculus test: questions

1 – Indefinite integral [2 pts]

First, find \(\int\left(8x^{3}-6x+\frac{1}{x}\right)dx\).

2 – Definite integral [2 pts]

Next, evaluate \(\int_{0}^{\pi/2}(\cos x+2)\,dx\).

3 – FTC with the chain rule [3 pts]

Then, 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)

4 – Accumulation from a graph [3 pts]

The graph of f.

Now, 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)

5 – Initial value problem [2 pts]

Also, find \(f(x)\) if \(f^{\prime}(x)=4x^{3}-3\) and \(f(2)=10\).

Question 6 and after

6 – Motion with integrals [3 pts]

Velocity in feet per second.

First, 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)

7 – Net change in context [3 pts]

Leak rate in gallons per hour.

Next, 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 and after

8 – Properties [2 pts]

Then, 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 fundamental theorem of calculus test

Key facts

In addition, FTC Part 1: d/dx of the integral from a to x of f(t) dt equals f(x).
Now, FTC Part 2: the integral from a to b of f equals F(b) – F(a) for any antiderivative F.
So, for an upper limit g(x), multiply by g′(x) (chain rule).
Then, every indefinite integral needs + C; an initial condition determines C.

Common mistakes with this fundamental theorem of calculus test

Also, Students differentiate the integral without the chain rule. Instead, substitute the limit into the integrand and multiply by the derivative of the limit.
Then, Students find the antiderivative but forget to use the initial condition. Without the constant, the answer is incomplete and loses a point.
Next, Students apply a property that does not exist, such as splitting the integral of a product. Only sums and constant multiples split.

Test-taking strategy

In addition, start with questions 1, 2, 5 and 8, which need short calculations and are worth 2 points each. Then work on the 3-point questions.
Now, Keep one line for the antiderivative and one for the evaluation. Separate lines make substitution errors easy to spot during a check.
So, in context questions, write a sentence with units at the end. Also, it confirms your answer is reasonable and satisfies the grader.

Also, Download the fundamental theorem of calculus test as a PDF with the button below and keep it with your notes.

Fundamental theorem of calculus test: frequently asked questions

Do I need to show the antiderivative on a definite integral question?

Then, Yes. Writing the antiderivative and the substitution earns partial credit if you make an arithmetic mistake. Also, a bare number with no work earns little, and a wrong number with no work earns nothing.

How do I answer a question that asks for the value of the accumulation function at a point?

Next, compute the signed area from the starting point to the given x using the graph. Treat regions above the axis as positive and below as negative. Next, add the pieces, using geometry for triangles and rectangles.

How do I use the properties of integrals?

In addition, you can reverse limits with a sign change, split an interval at a point, and pull constants out. Also, these facts let you combine given values. For instance, the integral from 0 to 5 equals the integral from 0 to 2 plus the one from 2 to 5.

So, all AP Calculus AB chapters: AP Calculus AB math.

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