AP Calculus AB review AP微积分复习提纲.docx

AP CALCULUS AB REVIEWChapter 2 Differentiation Definition of Tangent Line with Slop m If f is defined on an open interval containing c, and if the limit limx0yx= limx0fc+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 fx= limx0fc+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 *ddxsinx= cosx *ddxcosx= -sinx *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. *Rolles 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) = fb- 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 minimum2. 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*Newtons 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)undu= un+1n+1+ C, n-1 2)sinudu= -cosu+ C 3)cosu du= sinu+ C 4)1u du= lnu *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 abfxdx=Fb- F(a). *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 1b-a abf(x)dx.*The second fundamental theorem of calculus If f is continuous on an open internal I containing a, then, for every x in the interval, ddx axftdt=f(x).*Integration by Substitution*Integration of Even and Odd Functions 1) If f is an even function, thenabfxdx=2abf(x)dx. 2) If f is an odd function, thenabfxdx=0.*The Trapezoidal Rule Let f be continuous on a, b. The trapezoidal Rule for approximating abfxdx is given by abfxdx b-a2n fx0+2fx1+2fx2+2fxn-1 +fxnMoreover, a n , the right-hand side approachesabfxdx.*Simpsons Rule (n is even)Let f be continuous on a, b. Simpsons Rule for approximating abfxdx is abfxdxb-a3n fx0+4fx1+2fx2+4fx3+4fxn-1+fxnMoreover, as n, the right-hand side approaches abfxdx*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, gx= 1f(g(x) , f(g(x)0. *The Derivative of the Natural Exponential Function Let u be a differentiable function of x.1. ddxex= ex 2.ddxeu= eududx . *Integration Rules for Exponential Functions Let u be a differentiable function of x. eudu= eu+C. 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.ddxau=(lna)aududx 2.ddxlogau=1ulnadudx axdx=1lnaax+C limx(1+1x)x=limx(x+1x)x=e *Derivatives of Inverse Trigonometric Functions Let u be a differentiable function of x. ddxsin-1u=u1-u2 ddxcos-1u=-u1-u2 ddxtan-1u=u1+