AP Calculus AB review AP微积分复习提纲

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
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
provided the limit exists. For all x for which this limit exists, f’ is a function of x.
*The Power Rule
*The Product Rule
*
*
*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) = .
*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 HAVE TO KNOW IF U DO NOT
WANT TO SCORE 5)
Chapter 4 & 5
Integration
*Be able to solve a differential equation
*Basic Integration Rules
1)
2)
3)
4)
*Integral of a function is the area under the curve
*Riemann Sum (divide interval into a lot of sub-intervals, calculate the area for each sub-interval and summation is the
integral).
*Definite integral
*The Fundamental Theorem of Calculus
If a function f is continuous on the closed interval [a, b]
and F is an anti-derivative of f on the interval [a, b], then
.
*Definition of the Average Value of a Function on an Interval If f is integrable on the closed interval [a, b], then the average value of f on the interval is
.
*The second fundamental theorem of calculus
If f is continuous on an open internal I containing a, then,
for every x in the interval,
.
*Integration by Substitution
*Integration of Even and Odd Functions
1) If f is an even function, then.
2) If f is an odd function, then.
*The Trapezoidal Rule
Let f be continuous on [a, b]. The trapezoidal Rule for
approximating is given by
Moreover, a n →∞, the right-hand side approaches.
*Simpson’s Rule (n is even)
Let f be continuous on [a, b]. Simpson’s Rule for
approximating is
Moreover, as n→∞, the right-hand side approaches
*Inverse functions(y=f(x), switch y and x, solve for x)
*The Derivative of an Inverse Function
Let f be a function that is differentiable on an interval
I. If f has an inverse function g, then g is differentiable
at any x for which f’(g(x))≠0. Moreover,
, f’(g(x))≠0.
*The Derivative of the Natural Exponential Function
Let u be a differentiable function of x.
1. 2. .
*Integration Rules for Exponential Functions
Let u be a differentiable function of x.
.
♠Derivatives for Bases other than e
Let a be a positive real number (a ≠1) and let u be a differentiable function of x.
1. 2.
♠
♠
*Derivatives of Inverse Trigonometric Functions
Let u be a differentiable function of x.
*Definition of the Hyperbolic Functions。