The Fundamental Theorem of Calculus and Accumulation Functions
Differentiate an accumulation function using continuity, moving endpoints, and the chain rule.
College Board Topic 6.4 · CED p. 116 (PDF p. 121) · CED effective Fall 2020; project course outline 2026–27.
A(x)=∫ₐˣ f(t)dt is a function: the endpoint x moves, so the amount accumulated moves too. If f is continuous, the Fundamental Theorem of Calculus gives A′(x)=f(x). You are differentiating a growing signed area, and its instantaneous growth rate is the current height of the integrand. Also A(a)=0.
If the upper endpoint is u(x), apply the chain rule: d/dx ∫ₐᵘ⁽ˣ⁾ f(t)dt = f(u(x))u′(x). For two moving endpoints, subtract the lower endpoint contribution: f(u)u′−f(v)v′. Check continuity near the endpoints before using the theorem. An integral-defined function remains perfectly legitimate even when f has no elementary antiderivative.
Let's work through it
- Let A(x)=∫₀ˣ² cos(t)dt. First read the moving endpoint u(x)=x². Reason: The Fundamental Theorem differentiates with respect to the endpoint.
- A′(x)=cos(x²)·2x. Reason: Multiply by the endpoint derivative using the chain rule.
Try it from memory
Pause & retrieve
Try it before you peek.
Write your answer and the reason from memory. Then compare with the worked answer. Your self-assessment is a study decision, not an automatically graded score.
Question 1
If G(x)=∫₃ˣ e^(−t²)dt, what is G′(x)?
See every recall question with its answer
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Retrieval 1
If G(x)=∫₃ˣ e^(−t²)dt, what is G′(x)?
Worked answer and reason
e^(−x²).
The integrand is continuous and the upper endpoint derivative is 1.
Retrieval 2
What is G(3)?
Worked answer and reason
0.
The integration interval has zero length.
Catch the tempting mistake
- You must find a closed-form antiderivative to differentiate an accumulation function.
- The Fundamental Theorem gives its derivative directly when the integrand is continuous.
Calculator workspace and limits
For calculus, set the angle selector to Radians. This scientific calculator evaluates arithmetic and functions. It does not graph arbitrary functions, find numerical roots, evaluate numerical derivatives, or compute definite integrals. Those tasks need an approved graphing tool; the May 2027 exam permits approved handheld graphing calculators and Bluebook’s built-in Desmos on calculator-required parts. A graph or decimal estimate helps you check a conclusion; your setup, hypotheses, units and justification still do the explaining.
Scientific calculator
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Put it to work
- Unit 6 problem 6 · No calculator · analytic
- Unit 6 problem 7 · No calculator · analytic
- Unit 6 problem 20 · No calculator · analytic
For the teacher
- Let students drag the endpoint in an accumulation graph, then explain why slope equals height. Include a reversed endpoint example to expose the minus sign.
Glossary and deeper reading · 4 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Function
A mathematical function assigns exactly one output to each input in its domain. Inputs can share an output, but one input cannot have two different outputs. For example, f(x)=x² assigns the output 9 to both 3 and −3. Function notation names the rule and its input; f(3) means the output when the input is 3.
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Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus connects accumulation with differentiation. For a continuous integrand, differentiating its accumulation function returns the integrand. It also allows a definite integral of a continuous function to be evaluated by the endpoint change of an antiderivative. For example, the integral of 2x from 1 to 3 equals 3²−1²=8.
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Chain rule
The chain rule differentiates a composition by multiplying the outer derivative at the inner output by the inner derivative. For differentiable functions, (f(g(x)))′=f′(g(x))g′(x). For example, the derivative of sin(x²) is cos(x²)·2x. Nested compositions require a factor for each successive layer.
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Continuity
A function is continuous at an input when its value exists, its finite limit exists, and those two values agree. At a domain endpoint, continuity uses the relevant one-sided approach. For example, x² is continuous at 3 because both its value and its nearby limit are 9. Continuity on an interval requires the appropriate condition at every input in that interval.