Unit 6 Notes: Understanding the Fundamental Theorem of Calculus (AP Calculus BC)

FTC and Accumulation Functions

What an accumulation function is (and why you should care)

An accumulation function is a function built from an integral that “keeps track” of how much total change has built up from a starting point to a variable endpoint. The key idea is that definite integrals measure accumulated change (like total distance from velocity, or total mass from a density), and if you let the upper (or lower) limit be a variable, you get a new function whose output changes as that endpoint moves.

A standard accumulation function looks like this:

g(x) = \int_a^x f(t)\,dt

Here’s what each piece means:

• f(t) is the rate or density you are accumulating.
• a is the fixed starting point.
• x is the variable endpoint.
• g(x) is the accumulated net amount from a to x.

This matters because it links two big ideas in calculus:

• Differentiation: instantaneous rate of change
• Integration: accumulated change

The Fundamental Theorem of Calculus (FTC) is exactly the bridge between them.

The Fundamental Theorem of Calculus, Part 1 (FTC1)

FTC1 tells you how to differentiate an accumulation function.

If g(x) = \int_a^x f(t)\,dt and f is continuous, then

g'(x) = f(x)

Why this is true (conceptually)

Think about increasing x by a tiny amount \Delta x. The accumulation g(x) increases by approximately the area of a thin rectangle:

• width \Delta x
• height approximately f(x)

So the change is approximately f(x)\Delta x, and dividing by \Delta x gives a rate of change about f(x). Taking the limit makes that exact.

Differentiating more complicated accumulation functions

AP Calculus often uses variations where the variable shows up in places other than “upper limit equals x.” The core strategy is: use FTC1 plus the Chain Rule, and pay attention to orientation.

Case 1: Upper limit is a function of x

g(x) = \int_a^{h(x)} f(t)\,dt

Then

g'(x) = f(h(x))\cdot h'(x)

You’re doing FTC1 (replace x by h(x)), then multiplying by h'(x) because the endpoint is moving at rate h'(x).

Case 2: Lower limit is a function of x

g(x) = \int_{h(x)}^a f(t)\,dt

You can rewrite by reversing limits:

\int_{h(x)}^a f(t)\,dt = -\int_a^{h(x)} f(t)\,dt

So

g'(x) = -f(h(x))\cdot h'(x)

A common mistake is forgetting this negative sign when the variable is in the lower limit.

Case 3: Both limits depend on x

g(x) = \int_{u(x)}^{v(x)} f(t)\,dt

Rewrite as a difference:

g(x) = \int_a^{v(x)} f(t)\,dt - \int_a^{u(x)} f(t)\,dt

Differentiate:

g'(x) = f(v(x))\cdot v'(x) - f(u(x))\cdot u'(x)

Worked examples (FTC1)

Example 1: Basic accumulation derivative

Let

g(x) = \int_2^x \sqrt{t^2+1}\,dt

By FTC1,

g'(x) = \sqrt{x^2+1}

The dummy variable t becomes x after differentiating.

Example 2: Chain rule with an inside function

Let

g(x) = \int_0^{x^3} \cos(t)\,dt

Differentiate:

g'(x) = \cos(x^3)\cdot 3x^2

Example 3: Variable in the lower limit

Let

g(x) = \int_x^5 \ln(t)\,dt

Rewrite:

g(x) = -\int_5^x \ln(t)\,dt

Differentiate:

g'(x) = -\ln(x)

Notation reference (common FTC forms)

Exam Focus

• Typical question patterns:
  • Differentiate a function defined by an integral with variable limits, sometimes with a composite limit like x^2+1.
  • Find g'(c) at a point using a table/graph of f (you plug in, not integrate).
  • Write an expression for a new function as an accumulation function and then differentiate it.
• Common mistakes:
  • Forgetting the Chain Rule when the upper limit is h(x).
  • Dropping the negative sign when the variable is in the lower limit.
  • Confusing the dummy variable (like t) with the outside variable (like x).

Interpreting the Behavior of Accumulation Functions

Turning “area under a curve” into function behavior

When you define

g(x) = \int_a^x f(t)\,dt

you can learn how g behaves (increasing/decreasing, concavity, extrema) from the graph or sign of f—often without computing any integrals exactly.

The reason is FTC1 gives you an immediate relationship:

g'(x) = f(x)

So the derivative of the accumulation function is literally the original integrand.

Increasing and decreasing

A function increases where its derivative is positive. Since g'(x)=f(x):

• g is **increasing** where f(x) > 0.
• g is **decreasing** where f(x) < 0.

Interpretation-wise: if the “rate” f is positive, the accumulation is growing; if the “rate” is negative, the accumulation is shrinking.

Example 1: Monotonicity from a sign chart

Suppose a graph shows f(x) is positive on 1