Accumulation Functions

AP Calculus · Axiom Academy

Functions defined by integrals and the Second Fundamental Theorem Step 1: What Is an Accumulation Function? An accumulation function is a function defined by a definite integral whose upper limit is the variable: The output of is the net signed area under the curve from the fixed starting point to the variable endpoint . As moves to the right, the accumulated area changes: regions where add positive area, and regions where subtract area. Step 2: The Second Fundamental Theorem of Calculus Here is one of the most powerful results in all of calculus. If is continuous on , and we define: In plain language: the derivative of the accumulation function is the original integrand evaluated at . Differentiation undoes integration. This is the bridge that formally connects the two main operations of calculus. What if the upper limit is not just , but a function of ? We apply the chain rule: You evaluate the integrand at the upper limit , then multiply by the derivative of that upper limit . Step 4: Analyzing Accumulation Functions from a Graph On the AP exam, you are often given the graph of and asked to analyze . The key relationships: is increasing where (graph of is above the x-axis) is decreasing where (graph of is below the x-axis) has a local max or min where changes sign (graph crosses the x-axis) is concave up where is increasing; concave down where is decreasing has an inflection point where has a local max or min

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