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Chapter 3 Derivatives Section 1 Derivatives and Rates of Change 1

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1.1 Review of tangent line and instatanious velocity Definition The tangent line is the limit position of the secant line. 2

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Definition The derivative of a function f at a number a, denoted by is If this limit exists. 3

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Example 1 Find the Derivative of the function at the number a. Example 2 Find an equation of the tangent line to the parabola at the point (3,-6). 4

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1.2 Rates of change is called the average rate of change of y with respect to x over the interval 5

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1.3 The interpretation of the instantaneous rate of change The derivative is the instantaneous rate of change of y=f(x) with respect to x when x=a The derivative is large, the curve is steep, the y-values change rapidly. The derivative is small, the curve is flat, the y-values change slowly. 6

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Section 2 the Derivatives As a Function is called the derivative at a fixed number a. is called the derivative of f. Given any number x for which this limit exits, we assign x to the number 2.1 The derivative of a function 7

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Example 1 The graph of a function f is given in the following graph, use it to sketch the graph of the derivative 8

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2.2 Other notation Definition It is differentiable on an open interval (a,b) if it is differentiable at every number in the interval. 9

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Example 2 (a) If, find (b) Illustrate by comparing the graph of f and Example 3 If,find the derivative of f. State the domain of Example 4 Find if 10

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2.3 the relationship between continuous and differentiable Example 5 Where is the function differentiable? Theorem If f is differentiable at a, then f is continuous at a. Conclusion (1)The inverse proposition is not ture, that is if f is continuous at a, the function is not necessary differentiable at a. (2)But the inverse negative proposition is true, that is if f is not continuous at a, then f is not differentiable at a definitely. 11

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2.4 How can a function fail to be differentiable? Three ways for f not to be differentiable at a A cornerA discontinuityA vertical tangent 12

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2.5 Higher derivatives Definition If f is a differentiable function, and its derivative have derivative of its own, denoted by then this new function is called the second derivative of f. Example6 If,find 13

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Section 3 Differentiation Formulas Derivative of a constant function Derivative of a power function 14

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Differentiation rules Suppose f and g are differentiable functions, then their sum, difference, product and quotient are also differentiable functions, and we have The constant multiple rule If c is a constant and f is a derivative function, then 15

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Example 7 Find Example 8 Find the points on the curve where the tangent line is horizontal. Example 9 Find Example 10 Example 11 Let 16

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Section 4 Derivative of trigonometric functions Preparation 17

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Example 12 Find the 27 th derivative of Example 13 Find Example 14 Calculate 18

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Section 5 The Chain Rule The chain rule If g is differentiable at x and f is differentiable at g(x), then the composite function F(x)=f(g(x)) is differentiable at x and is given by the product That means, if y=f(u) and u=g(x) are both differentiable function, then When we use this formula we should bear in mind dy/dx refers to the derivative of y with respect to x, (i.e. y should be regarded as a function of x). dy/du refers to the derivative of y with respect to u, (i.e. y should be regarded as a function of u). 19

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Example 15 Suppose y=sin2x, find dy/dx. Example 16 Differentiate Example 17 Differentiate Example 18 Differentiate Example 19 Differentiate 20

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The power rule (general version) If n is any number, then Example 20 Differentiate the following functions. Example 21 Find equation of the tangent line and normal line to the curve at the point (1,1/2). 21

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Section 6 Implicit Function So far,we have met function can be discribed byexpressing one variable explicitly in terms of another variable. However, some functions are defined implicitly by a relation between x and y such as How to find the derivative of y without solving an equation y in terms of x. Folium of Decartes. 22

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Implicit Differentiation Step 1. Differentiate both side of the equation with respect to x. Step 2. Solve the resulting equation for Example 1. (a) (b) Find an equation of the tangent to the circle 23

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Example 2. (a) (b) Find the equation of the tangent to the folium of Decartes at the point (3,3). (c) At what point in the first quadrant is the tangent line horizontal? Example 3. Find Example 4. Find 24

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Section 9 Linear Approaximations and Differentials If f is differentiable at a, then when we zoom in toward the point (a,f(a)) The graph straightens out and appears more and more like a line. This observation is the basis for a method of finding approximate values of functions 25

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That means we can use the tangent line at (a,f(a)) as an approximation to the curve y=f(x) when x is near a. (a,f(a)) linear approximation of f at a or tangent line approximation linearization of f at a 26

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Example 5 Find the linearization of the function at a=1, and use it to approximate the numbers Are these approximations overestimates or underestimates? xFrom L(x)Actual value 27

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Differential the ideas behind the linear approximation is differential. We can use the value of the function at x plus dy to approximate the value of the function at Because dy is a linear function, it is easier to calculate than. dy it is called the differential 28

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Example 6 Compare the values of and x changes (a) from 2 to 2.05 (b) from 2 to 2.01 Example 7 Using the linear approximation to estimate tan44 。 29

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Example 5 The radius of sphere was measured and found to be 21cm, with the possible error in measurement of at most 0.05cm. Find the approximation to the volume and the relative error. 30

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3 DERIVATIVES. The functions that we have met so far can be described by expressing one variable explicitly in terms of another variable. For example,,

3 DERIVATIVES. The functions that we have met so far can be described by expressing one variable explicitly in terms of another variable. For example,,

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