AP Calculus Class Notes:
Limits:
Definition: The limit of a function as x approaches a certain value is the value that the
function approaches as x gets closer and closer to that value.
Notation: limit as x approaches a of f(x) = L
Examples:
limit as x approaches 2 of x^2 - 4x + 4 = 4
limit as x approaches 0 of sin(x)/x = 1
Derivatives:
Definition: The derivative of a function at a certain point is the slope of the tangent line to
the function at that point.
Notation: f'(x) or dy/dx
Examples:
f(x) = x^2, f'(x) = 2x
f(x) = e^x, f'(x) = e^x
Integrals:
Definition: The definite integral of a function from a to b is the area between the graph of
the function and the x-axis, between the points x = a and x = b.
Notation: ∫f(x)dx from a to b
Examples:
∫x^2dx from 0 to 1 = 1/3
∫e^xdx from 0 to 1 = e^1 - e^0 = e - 1
The Fundamental Theorem of Calculus (FTC):
The FTC connects the concept of derivatives and integrals by stating that the derivative of
the definite integral of a function is the original function.
Notation: ∫f(x)dx = F(x) + C, where F(x) is the antiderivative of f(x) and C is a constant.
Examples:
∫(2x)dx = x^2 + C
∫(e^x)dx = e^x + C
Implicit differentiation:
Implicit differentiation is a method of finding the derivative of an equation where y is not
explicitly defined as a function of x.
Notation: dy/dx = ...
Example:
x^2 + y^2 = 4, dy/dx = -x/y.
Taylor Series:
Taylor series are a representation of a function as an infinite sum of terms calculated from
the values of the function’s derivatives at a single point.
Notation: Tn(x) = f(a) + (f'(a))(x-a) + (f''(a))(x-a)^2/2! + ... + (f^n(a))(x-a)^n/n!
