微積分英文版教學(xué)課件：Chapter2 Derivatives

1、Chapter2 Derivatives2.1 The Derivative as a function The Tangent Problem Let f be a function and let P(a, f(a) be a point on the graph of f. To find the slope m of the tangent line l at P(a, f(a) on the graph of f, we first choose another nearby point Q(x, f(x) on the graph (see Figure 1) and then c

2、ompute the slope mPQ of the secant line PQ.0P(a,f(a)Q(x,f(x)f(x)0P(a,f(a)Q(x,f(x)f(x)lLet Q get closer to P and QP. The slope m of the tangent line l is the limit of the slopes of the secant lines,i.eLet Then So the slope of the second line PQ is The slope m of the tangent line l isThe velocity prob

3、lemSuppose an object moves along a straight line according to an equation of motion is called the position function of the objectosAverage velocity Rates of changeIf x change from to ,then then change , The corresponding change in y isThe instantaneous velocity at t=a then the change in (increment o

4、f )isThe average rate of change of y with respect to xisThe instantaneous rate of change of y with respect to x at isDefinition of DerivativeDefinition Let y=f(x) be a function defined on an open interval containing a number a.The derivative of f(x) at number a, denoted by f(a) , is if this limit ex

5、ists. If we write x=a+h, then h=x-a and h approaches 0 if and only if x approaches a.thenThe right-hand derivative of f at a ,is denoted by f +(a)The left-hand derivative of f at a , is denoted by f -(a).f (a) exists if and only if both the right-hand derivative f +(a) and the left-hand derivative f

6、 -(a) exist and are equal. If exists, we say that f(x) is differentiable at a or that f(x) has a derivative at a. Example Let f(x)=x.Show that f (x) is not differentiable at 0.Proof By Definition, we have Therefore the function f(x) is not differentiable at 0.Interpretation of the Derivative as the

7、Slope of a TangentThe tangent line to at is the line through whose slope is equal to ,the derivative of at a.0P(a, f(a)Q(x, f(x) f(x)lAn equation of the tangent line to the curve at the point : The geometric interpretation of a derivative.Example Find an equation of the tangent lineto the parabola a

8、t the point (4,2). SolutionThe slope of the tangent line at (4,2) isan equation of the tangent line isorInterpretation of the Derivative as a Rate of ChangeThe derivative is the instantaneous rate of changeof with respect to when Example Find and given that Solution By definition Now lets calculate

9、Since f is not defined by the same formula on both sides of 1, we will evaluate this limit by taking one-sided limits. To the left of 1, Thus To the right of 1, ThusAnd is a function of x, denoted by Leibniz notationA function f is differentiable on an open interval I if f (x) exists for every x in

10、that interval I. ThenSoExample Find the derivative of Solutionso A function f is differentiable on a closed interval a, b if f is differentiable on an open interval (a, b) and both the right-hand derivative f +(a) and the left-hand derivative f -(b) existTheorem If a function f is differentiable at

11、a number, then it is continuous at a. Proof we have Hence Therefore This implies that f is continuous at a. NOTE 1.The converse of theorem is false 2. f is not continuous at a, then f is notdifferentiable at a.Three ways for f(x) not to be differentiable at a 1. a corner at a 2. discontinuous at a 3

12、. have a vertical tangent line at aaaaTechniques of Calculating DerivativesEx1. (C is a constant). Solution:That is Ex 2. solution:Find Note：For any power function eg，（we will prove it later!）Ex 3. . solution:That is, Similarly:Exponential Functionsh0.10.010.0010.00010.71770.69560.69340.69321.16121.

13、10471.09921.0987Definition of the Number ee is a number such that Exponential FunctionsDerivative of the natural exponential FunctionDifferential Laws Suppose that u(x) and v(x) are differentiable, then their sum, difference, product, and quotient are also differentiable, andPf: Let, thenThe law can

14、 be generalized to more functions.(2)Pf: LetthenCorollary:( C is a constant )Ex 1. Solution:(3)Pf: LetthenCorollary:( C is a constant )Example:Pf: Similarly:Laws for inverse function:Th2. is monotonic and differentiable, Pf:Ex3. Derivatives for inverse trigonometric functions.1) LetthenSimilarly:using, thenEx4:Find the derivative of The chai