AP CALCULUS AB REVIEW
Chapter 2
Differentiation
Definition of Tangent Line with Slop m
If f is defined on an open interval containing c, and if the limit
lim ∆x→0∆y
∆x
=lim
∆x→0
f(c+∆x)−f(c)
∆x
=m
exists, then the line passing through (c, f(c)) with slope m is the tangent line to the graph of f at the point (c, f(c)).
Definition of the Derivative of a Function
The Derivative of f at x is given by
f′(x)=lim
∆x→0f(c+∆x)−f(c)
∆x
provided the limit exists. For all x for which this limit exists, f’is a
function of x.
*The Power Rule
*The Product Rule
*d
dx
[sin x]=cos x
*d
dx
[cos x]=−sin x
*The Chain Rule
☺Implicit Differentiation (take the derivative on both sides; derivative
of y is y*y’)
Chapter 3
Applications of Differentiation
*Extrema and the first derivative test (minimum: − → + , maximum: +
→ −, + & − are the sign of f’(x) )
*Definition of a Critical Number
Let f be defined at c. If f’(c) = 0 OR IF F IS NOT DIFFERENTIABLE
AT C, then c is a critical number of f.
*Rolle’s Theorem
If f is differentiable on the open interval (a, b) and f (a) = f (b), then there
is at least one number c in (a, b) such that f’(c) = 0.
*The Mean Value Theorem
If f is continuous on the closed interval [a, b] and differentiable on the
open interval (a, b), then there exists a number c in (a, b) such that f’(c) =
f(b)− f(a)
.
b−a
*Increasing and decreasing interval of functions (take the first derivative)
*Concavity (on the interval which f’’ > 0, concave up)
*Second Derivative Test
Let f be a function such that f’(c) = 0 and the second derivative of f exists
on an open interval containing c.
1.If f’’(c) > 0, then f(c) is a minimum
2.If f’’(c) < 0, then f(c) is a maximum
*Points of Inflection (take second derivative and set it equal to 0, solve the
equation to get x and plug x value in original function)
*Asymptotes (horizontal and vertical)
*Limits at Infinity
*Curve Sketching (take first and second derivative, make sure all the
characteristics of a function are clear)
♫ Optimization Problems
*Newton’s Method (used to approximate the zeros of a function, which is
tedious and stupid, DO NOT HA VE TO KNOW IF U DO NOT WANT
TO SCORE 5)
Chapter 4 & 5
Integration
*Be able to solve a differential equation
*Basic Integration Rules
