The Fundamental Theorem of Calculus, Part 1, connects differentiation and integration by showing how accumulation functions behave and how their derivatives reveal underlying instantaneous rates.

Understanding Accumulation Functions Defined by Integrals

An accumulation function represents the total accumulated change of a quantity from a fixed starting point up to a variable upper limit. In this subsubtopic, the focus is on how to differentiate such functions when they are expressed as definite integrals with a variable bound. This idea forms the core of the Fundamental Theorem of Calculus, Part 1 (FTC Part 1), which provides a powerful link between rate functions and the accumulation of change over an interval.

When a function is defined using a definite integral whose upper limit is a variable, the function’s derivative can often be found without integrating at all. Instead, FTC Part 1 states that the derivative of the accumulation function equals the original integrand evaluated at the variable upper limit, provided the integrand is continuous on the interval.

FTC Part 1 and Its Meaning

FTC Part 1 tells us how accumulation behaves instantaneously. Suppose an accumulation function increases over an interval by summing contributions from a rate function. Then its derivative measures the instantaneous rate of accumulation — precisely the value of the rate at that point. This relationship is foundational in AP Calculus AB because it allows students to transition between integrals as totals and derivatives as rates.

Before using FTC Part 1, students must understand how integrals define new functions and how the structure of the integral affects differentiation.

Key Structure of an Accumulation Function

A typical accumulation function has the following form:

• A constant lower limit, representing the initial point of accumulation.

• A variable upper limit, usually written as $x$.

• A continuous integrand, representing the rate of change being accumulated.

Because the upper limit is variable, changes in $x$ alter the accumulated total, making $g$ behave like any differentiable function whose derivative depends on the rate function $f$

The shaded region represents the accumulated area under a continuous function from $x = a$ to a variable endpoint $x$. As $x$ moves, the value of the accumulation function $F(x) = \int_a^x f(t), dt$ changes with the size of this region. This image directly supports interpreting accumulation functions as “area from a fixed start to a moving boundary.” Source.

A student must be aware that the integrand uses $t$ as the variable of integration, which is independent of $x$. This separation allows $x$ to serve as a boundary while the integral itself accumulates values of $f(t)$.

Applying the Fundamental Theorem of Calculus, Part 1

FTC Part 1 directly describes the derivative of an accumulation function.

This result means that a definite integral does more than compute total change; it also constructs a function whose derivative retrieves the original rate. For AP students, this provides a crucial bridge between the two major branches of calculus.

When applying this relationship, it is essential that the integrand be continuous on the interval of interest. Continuity guarantees that the accumulation process is well-behaved and differentiable at every point in the domain.

How Differentiation Works in This Context

Differentiating a function given as an integral does not involve traditional differentiation rules until the structure of the integral changes. In its simplest form — constant lower limit, variable upper limit, no transformations applied to the limit — the differentiation becomes immediate through FTC Part 1.

Students should observe that the upper limit of integration determines which value of the integrand becomes the derivative. Conceptually, the small increase in accumulated area as $x$ grows corresponds exactly to the height of the graph of $f$ at $x$. Thus, a thin vertical slice added at $x$ represents the instantaneous rate, reinforcing the geometric intuition behind FTC Part 1.

The blue and red shaded regions show the accumulated area up to $x$ and the extra area added from $x$ to $x+h$. The expressions $A(x+h)-A(x)$ and $h \cdot f(x)$ compare the true additional area with the area of a thin rectangle whose height is the function value at $x$. Although the formulas and the fraction $\dfrac{A(x+h)-A(x)}{h}$ go slightly beyond the AP statement, they concretely illustrate how the derivative of the area function approaches $f(x)$ as $h$ becomes small. Source.

Variations and Extensions within AP Calculus AB

While full generalizations are explored in higher mathematics, AP Calculus AB focuses on understanding the fundamental structure:

• The lower limit must be constant.

• The variable upper limit must be the differentiation variable.

• The integrand must be continuous.

Under these conditions, students apply FTC Part 1 directly, even when the integrand is algebraic, trigonometric, exponential, or a combination of these.

Occasionally, the upper limit may involve a more complex expression. Although FTC Part 1 still applies, such cases typically require an additional chain rule step. These instances remain aligned with the same essential idea: the derivative of an accumulated quantity reflects the rate at the boundary of accumulation.

Interpreting FTC Part 1 in Context

In applied problems, the accumulation function often represents a total amount — distance, volume, mass, or another physical quantity — derived from a rate function. FTC Part 1 connects this total to the instantaneous rate. When applying the theorem, students learn to:

• Identify the integrand as the underlying rate of change.

• Recognize the accumulation function as representing a total amount or net change.

• Understand that the derivative of this total retrieves the instantaneous rate at the upper limit.

These ideas allow students to interpret accumulation functions fluently using graphical, numerical, or verbal descriptions of the rate function.

Summary of Student Responsibilities for This Subsubtopic

Students working with FTC Part 1 should be able to:

• Explain why the derivative of an accumulation function equals the integrand evaluated at the upper limit.

• Recognize functions defined by integrals and determine when FTC Part 1 applies.

• Use proper interpretation of variables in the integrand versus the variable limit.

• Apply differentiation rules when the structure of the upper limit requires additional steps.

These skills form an essential part of understanding the relationship between total change and instantaneous rate within AP Calculus AB.