1、A DERIVATIVE FUNCTION1. The derivative function or simply the derivative is defined as 2. Find the derivative functiona) Find ,b) Find the average rate of change ,c) Find the limit .3. Geometric significanceConsider a general function y=f(x), a fixed point A(a,f(a) and a variable point B(x,f(x). The
2、 slope of chord AB=. Now as BA, xa and the slope of chord ABslope of tangent at A.So, is .Thus, we can know the derivative at x=a is the slope of the tangent at x=a.4. RulesC(a constant)0 5. The chain ruleIf where then . , 6. Inverse function, Parametric function and Implicit functionInverse functio
3、n:, , i.e., , Parametric function:, i.e., , Implicit function: , ., , t7. High derivativey=sinx , B APPLICATIONS OF DIFFERENTIAL CALCULUS1. Monotonicitya) If S is an interval of real numbers and f(x) is defined for all x in S, then： f(x) is increasing on S for all x in S, and f(x) is decreasing on S
4、 for all x in S.b) Find the monotone intervall Find domain of the function,l Find , and x which make ,l Draw sign diagram, find the monotone interval.2. Maxima/Minima, Horizontal inflection, Stationary pointC INTEGRAL1. The idea of definite integralWe define the unique number between all lower and u
5、pper sums as and call it “the definite integral of from a to b”,i.e., where .We note that as , and We write .If for all x on a,b then is the shaded area.2. Properties of definite integrals, c is any constant, where （f(x) odd）,(f(x)even)If on then If on then The average value of a function on an interval a,b3. The infinite integralIf , then Formulas:, , (), U Substitution substitution u=g(x) Integration by Parts
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