APCalculusABreviewAP微积分复习提纲

1、f(x) =limf(cf(c + + Ax)Ax) - - f(c)f(c)AxAxAP CALCULUS AB REVIEWChapter 2Differe ntiatio nDefinition of Tangent Line with Slop mIf f is defi ned on an ope n in terval containing c, and if the limitAyAyf(cf(c + + AxJAxJ - - f(c)f(c)1*1*II +lim = lim -= mexists, then the line passing through (c, f(c)

2、with slope m is the tangent line to the graph of f at the point (c, f(c)、Defin iti on of the Derivative of a Fun cti on The Derivative of f at x is given byprovided the limit exists 、 For all x for which this limit exists, f is afun cti on of x、*The Cha in Rule? Implicit Differentiation (take the de

3、rivative on both sides; derivative of y is y*y )Chapter 3Applicati ons of Differe ntiatio n*Extrema and the first derivative test (minimum:- 宀 + , maximum: + f - , + & - are the sign of f ) (x)*Defi niti on of a Critical NumberLet f be defined at c、 If f (c) = 0 OR IF F IS NOT DIFFERENTIABLEAT C, th

4、e n c is a critical nu mber of f、*Rolle s TheoremIf f is differe ntiable on the ope n in terval (a, b) and f (a) = f (b), the n there is at least one nu mber c in (a, b) such thaf (c) = 0、*The Mean Value Theorem*The Power Rule *The Product RuleIf f is continuous on the closed interval a, b and diffe

5、rentiable on the狀4 11)l-I+ C, n 工一 1ope n in terval (a, b), the n there exists a nu mber c in (a, b) such that (c)=fW - - /(a)/(a)-、*ln creas ing and decreas ing in terval of fun cti ons (take the first derivative) *Con cavity (on the in terval which f( 0,c on cave up)*Sec ond Derivative TestLet f b

6、e a function such thatf (c) = 0 and the second derivative of f exists on an ope n in terval containing c1. 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 equati on to get x and plug x value in origi

7、nal fun cti on)*Asymptotes (horiz on tal and vertical)*Limits at Infinity*Curve Sketching (take first and second derivative, make sure all the characteristics of a fun cti on are clear)? Optimization Problems*Newton s Method (used to approximate the zeros of a function, which is tedious and stupid,

8、DO NOT HA VE TO KNOW IF U DO NOT WANT TO SCORE 5)Chapter 4 & 5In tegrati on*Be able to solve a differe ntial equati on*Basic In tegrati on Rules2)1引锂止餐；二一二亡电山匚3)1 n人仁i i ：.冷丨匚*ln tegral of a fun cti on is the area un der the curve*Riemann Sum (divide interval into a lot of sub-intervals, calculate t

9、hearea for each sub-i nterval and summati on is the in tegral、*Defi nite in tegral*The Fun dame ntal Theorem of CalculusIf a function f is continuous on the closed interval a, b and F is an an ti-derivative of f on the in terval a, b, the nJy(x)dx = 0Ja)dx = F(h)- F)、*Defi niti on of the Average Val

10、ue of a Fun cti on on an In tervalIf f is in tegrable on the closed in terval a, b, the n the average value of f on the in terval is*The sec ond fun dame ntal theorem of calculusIf f is con ti nu ous on an ope n intern al I containing a, the n, for every x in the in terval,即了 (t)处卜他、*ln tegrati on b

11、y Substituti on*Integration of Even and Odd Functions1) If f is an evenfunction, then 、2) If f is an odd fun cti on, the n*The Trapezoidal RuleLet f be continuous on a, b、 The trapezoidal Rule forMoreover, a n 卩 the right-ha nd side approaches、*Simps on Rule (n is eve n)Let f be con ti nu ous on a,

12、b、Simps on Rule for approximati ngisMoreover, as n, the right-ha nd side approaches*Inverse functions(y= f(x), switch y and x, solve for x)*The Derivative of an In verse Fun cti onLet f be a function that is differentiable on an interval I、 If f has an in verse fun cti on g, the n g is differe ntiab

13、le at any x for which f approximat ingis give n byg(x)和、Moreover,f g(x)和、1gW =*The Derivative of the Natural Exponen tial Fun cti on*ln tegrati on Rules for Exp onen tial Fun cti onsLet u be a differe ntiable fun cti on of x、?Derivatives for Bases other tha neLet a be a positive real nu mber (a 勺)and let u be a differe ntiablefun cti on of x、d r “I“Id d r ri i 1 1 JuJu.-r-a= (In a)a r- -log ul GJ?lim (1 += I im e?*Derivatives of In verse Trigo no metric Fun c