AP Calculus AB Fundamental Theorem of Calculus and Antiderivatives Lesson (Free PDF)

AP Calculus AB math lesson: Fundamental Theorem of Calculus and Antiderivatives – Maths PDF

Cover of the book AP Calculus AB Math Lessons (PDF)

Download as a PDFThe AP Calculus AB math lessons bookPrintable PDFSee the book ›

Sommaire

This ftc lesson 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.

The Fundamental Theorem of Calculus connects derivatives and integrals. Part 1 says the derivative of the accumulation function from a to x equals f(x). Part 2 says a definite integral equals F(b) – F(a), where F is any antiderivative of f. An indefinite integral adds the constant C, found from an initial condition.

AP Calculus AB · Common Core State 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

Key facts in this ftc lesson

Key facts

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.

What is an accumulation function?

Define \(F(x)=\int_{a}^{x}f(t)\,dt\). It records the area (signed) accumulated between \(a\) and \(x\). As \(x\) moves right, the area grows at a rate equal to the height of the graph.

F(x) is the area accumulated up to x; its rate of growth is the height f(x).

Fundamental Theorem of Calculus

Part 1: If \(f\) is continuous, \(\frac{d}{dx}\int_{a}^{x}f(t)\,dt=f(x)\).
Part 2: If \(F^{\prime}=f\), then \(\int_{a}^{b}f(x)\,dx=F(b)-F(a)\).

The two parts of the Fundamental Theorem link derivatives and integrals.

When the upper limit is a function \(g(x)\), use the chain rule: \(\frac{d}{dx}\int_{a}^{g(x)}f(t)\,dt=f(g(x))\cdot g^{\prime}(x)\). For example, \(\frac{d}{dx}\int_{0}^{x^{2}}\cos t\,dt=2x\cos(x^{2})\).

What is an antiderivative?

An antiderivative of \(f\) is a function \(F\) with \(F^{\prime}=f\). The family of all antiderivatives is the indefinite integral \(\int f(x)\,dx=F(x)+C\). The constant \(C\) appears because constants have derivative zero.

Basic antiderivatives to memorize (add + C for indefinite integrals).

How do you evaluate a definite integral?

Find an antiderivative \(F\) (no constant needed).
Evaluate \(F(b)-F(a)\).
Interpret the sign and units.

Example. \(\int_{1}^{3}(2x+1)\,dx=\left[x^{2}+x\right]_{1}^{3}=(9+3)-(1+1)=10\). Notice that the graph of \(F(x)=\frac{x^{3}}{3}-4x\) (green) falls while \(f(x)=x^{2}-4\) (blue) is negative.

F decreases while f is negative and increases when f is positive; F has its minimum where f changes sign.

The theorem also explains net change: if \(r(t)\) is a rate, then \(\int_{a}^{b}r(t)\,dt\) is the total change in the quantity from time \(a\) to time \(b\). A tank with inflow rate \(12-2t\) gallons per minute gains \(\int_{0}^{6}(12-2t)\,dt=36\) gallons in 6 minutes, and the answer carries the units gallons because (gallons per minute) times (minutes) is gallons.

Do not forget the constant of integration for an indefinite integral.
For a definite integral, write the antiderivative in brackets and subtract the lower value from the upper value.
If the graph crosses the axis, the integral is net (signed) area; split the interval to find total area.

What is an initial value problem?

If you know \(f^{\prime}(x)\) and a single value of \(f\), integrate to get \(f(x)+C\) and use the value to find \(C\). For \(f^{\prime}(x)=6x^{2}-2\) and \(f(1)=5\): \(f(x)=2x^{3}-2x+C\), and \(0+C=5\) gives \(C=5\).

Ftc lesson: questions and answers

What does the Fundamental Theorem of Calculus say?

It links differentiation and integration: the derivative of the accumulation function equals the integrand, and a definite integral equals the change in any antiderivative between the limits.

How do you find a derivative of an integral with a variable upper limit?

Substitute the upper limit into the integrand and multiply by the derivative of the upper limit. For the integral from 0 to x squared of cos t dt, the derivative is cos(x^2) times 2x.

Why do indefinite integrals include + C?

Functions that differ by a constant have the same derivative, so an antiderivative is determined only up to a constant. The + C represents that family.

What is the antiderivative of 1/x?

It is the natural logarithm of the absolute value of x, ln|x| + C. It is the missing case in the power rule for integrals.

How do you solve an initial value problem?

Find the general antiderivative with + C, substitute the given point to solve for C, and then write the particular function.

How to use this ftc lesson

Read the definitions first, then follow the worked method line by line and redo the example on paper without looking. When every step makes sense, move on to the worksheet, then check yourself with the quiz.

Download the ftc lesson as a PDF with the button below and keep it with your notes.

All AP Calculus AB chapters: AP Calculus AB math.

Voter.. post

Free printable PDF