APCalculusABreviewAP微积分复习提纲

1、精品文档AP CALCULUS AB REVIEWChapter 2DifferentiationDefinition of Tangent Line with Slop mIffis defined on an open interval containing c, and if the limitexists, 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

2、of a FunctionThe Derivative of f at x is given byprovided 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; derivativeof y is y*y)Chapter 3Applications of Differe

3、ntiation*Extrema and the first derivative test (minimum: - + , maximum: + - , + & - are the sign of f (x) )*Definition of a Critical Number.精品文档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 TheoremIf f is differentiable on the open

4、interval (a, b) and f (a) = f (b), then there is at least one number c in (a, b) such thatf (c) = 0.*The Mean Value TheoremIf 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 thatf (c) =.*Increasing and decreasing

5、 interval of functions (take the first derivative)*Concavity (on the interval which f 0, concave up)*Second Derivative TestLet 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 minimum2. If f (c) 0, then f(c) is a

6、 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 a

7、re 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 WANTTO SCORE 5)Chapter 4 & 5Integration*Be able to solve a differential equation.精品文档*Basic Integration Rules1)2)3)4)*Integral of a function is

8、 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 CalculusIf a function f is continuous on the closed interval a, b and F is an anti-derivative of f

9、 on the interval a, b, then.*Definition of the Average Value of a Function on an IntervalIf f is integrable on the closed interval a, b, then the average value of f on the interval is.*The second fundamental theorem of calculusIf f is continuous on an open internal I containing a, then, for every x

10、in the interval,.*Integration by Substitution.精品文档*Integration of Even and Odd Functions1) If f is an even function, then.2) If f is an odd function, then.*The Trapezoidal RuleLetfbecontinuousona,b.ThetrapezoidalRuleforapproximatingis given byMoreover, a n , the right-hand side approaches.*Simpsons

11、Rule (n is even)Letf be continuous on a, b. Simpsons Rule forapproximatingisMoreover, as n, the right-hand side approaches*Inverse functions(y= f(x), switch y and x, solve for x)*The Derivative of an Inverse FunctionLet f be a function that is differentiable on an interval I. If f has an inverse fun

12、ction 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.精品文档2.*Integration Rules for Exponential FunctionsLet u be a differentiable function of x.?Derivatives for Bases other than eLet a be a positive real n