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  2. AP Calculus
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Glossary

A

Accumulation Function

Criticality: 2

A function defined by a definite integral where one of the limits of integration is a variable, representing the accumulated area under a curve from a fixed point to the variable limit.

Example:

The function A(x)=∫0xe−t2dtA(x) = \int_0^x e^{-t^2} dtA(x)=∫0x​e−t2dt is an accumulation function that describes the area under the curve y=e−t2y=e^{-t^2}y=e−t2 from 0 to xxx.

C

Chain Rule (as applied to FTC)

Criticality: 3

A differentiation rule used when the upper limit of an integral is a function of $x$ (not just $x$). It requires multiplying the result of the FTC by the derivative of the upper limit function.

Example:

To find the derivative of F(x)=∫0tan⁡(x)t2+1dtF(x) = \int_0^{\tan(x)} \sqrt{t^2+1} dtF(x)=∫0tan(x)​t2+1​dt, you must use the Chain Rule, resulting in F′(x)=(tan⁡(x))2+1⋅sec⁡2(x)F'(x) = \sqrt{(\tan(x))^2+1} \cdot \sec^2(x)F′(x)=(tan(x))2+1​⋅sec2(x).

F

Fundamental Theorem of Calculus (FTC)

Criticality: 3

A foundational theorem that connects differentiation and integration, stating that the derivative of an integral with a variable upper limit is the function inside the integral, evaluated at that limit.

Example:

If you have G(x)=∫2xcos⁡(t)dtG(x) = \int_2^x \cos(t) dtG(x)=∫2x​cos(t)dt, then by the Fundamental Theorem of Calculus, G′(x)=cos⁡(x)G'(x) = \cos(x)G′(x)=cos(x).

I

Integral Rules (for switching bounds)

Criticality: 2

Properties of definite integrals, such as the rule that switching the limits of integration changes the sign of the integral. This is crucial when the variable is in the lower bound for FTC applications.

Example:

If you need to differentiate ∫x5f(t)dt\int_x^5 f(t) dt∫x5​f(t)dt, you first apply the integral rule ∫abf(x)dx=−∫baf(x)dx\int_a^b f(x) dx = -\int_b^a f(x) dx∫ab​f(x)dx=−∫ba​f(x)dx to rewrite it as −∫5xf(t)dt-\int_5^x f(t) dt−∫5x​f(t)dt before applying FTC.

U

Upper Bound (FTC with Chain Rule)

Criticality: 3

The upper limit of integration in a definite integral. When this bound is a function of $x$ (e.g., $x^2$), the Chain Rule must be applied when differentiating the integral using the FTC.

Example:

In H(x)=∫1x3sin⁡(t)dtH(x) = \int_1^{x^3} \sin(t) dtH(x)=∫1x3​sin(t)dt, the upper bound is x3x^3x3, so H′(x)=sin⁡(x3)⋅3x2H'(x) = \sin(x^3) \cdot 3x^2H′(x)=sin(x3)⋅3x2.