How does the fundamental theorem of calculus connect accumulation and rate of change?

Conditions

• The function $f$ is continuous.
• $F$ is an antiderivative of $f$ (i.e., $F' = f$).

Reasoning, step by step

1. Define the accumulation function $A(x) = \int_a^x f(t) dt$.
2. Recognize that $A'(x) = f(x)$ by the first part of the theorem.
3. Identify that if $F'(x) = f(x)$, then $A(x)$ and $F(x)$ differ by a constant.
4. Use the initial condition $A(a) = 0$ to determine the constant.
5. Conclude that $A(x) = F(x) - F(a)$, linking the definite integral to antiderivatives.

Example

The script states: 'The fundamental theorem relates accumulation and rate of change. If F′=f then $A(x)=F(x)-F(a)$. Antiderivatives may differ by constants. For accumulation of x² from zero, $A(0)=0$ selects $A(x)=x$³/3 from the family x³/3+C.'

Common misconceptions

• Believing that antiderivatives are unique without specifying a constant.
• Confusing the indefinite integral (family of functions) with the definite integral (specific value).

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Connected concepts

Explore next

• How is the derivative defined as a limit of a difference quotient?
• What is the geometric interpretation of the accumulation function?

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