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Chapter 3 Introduction to the Derivative Sections 3.5, 3.6, 4.1 and 4.2

Chapter 3 Introduction to the Derivative Sections 3.5, 3.6, 4.1 and 4.2. Introduction to the Derivative. Average Rate of Change The Derivative. Average Rate of Change. The change of f ( x ) over the interval [ a , b ] is.

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Chapter 3 Introduction to the Derivative Sections 3.5, 3.6, 4.1 and 4.2

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  1. Chapter 3Introduction to the DerivativeSections 3.5, 3.6, 4.1 and 4.2

  2. Introduction to the Derivative • Average Rate of Change • The Derivative

  3. Average Rate of Change The change of f(x) over the interval [a,b] is The average rate of change of f(x) over the interval [a,b] is Difference Quotient

  4. Average Rate of Change Is equal to the slope of the secant line through the points (a, f (a)) and (b, f (b)) on the graph of f (x) secant line has slope mS

  5. Average Rate of Change as h0 The average rate of change of f over the interval [a,b] can be written in two different ways:

  6. Average Rate of Change as h0 We now look at the behavior of the average rate of change of f(x) as b ah gets smaller and smaller, that is, we will let h tend to 0 (h0) and look for a geometric interpretation of the result. For this, we consider an example of values of and their corresponding secant lines.

  7. Average Rate of Change as h0 Secant line tends to become the Tangent line

  8. Average Rate of Change as h0 Secant line tends to become the Tangent line

  9. Average Rate of Change as h0 Secant line tends to become the Tangent line

  10. Average Rate of Change as h0 Secant line tends to become the Tangent line

  11. Average Rate of Change as h0 Secant line tends to become the Tangent line

  12. Average Rate of Change as h0 Secant line tends to become the Tangent line

  13. Average Rate of Change as h0 Secant line tends to become the Tangent line Zoom in

  14. Average Rate of Change as h0 Secant line tends to become the Tangent line

  15. Average Rate of Change as h0 Secant line tends to become the Tangent line

  16. Average Rate of Change as h0 • We observe that as h approaches zero, • The secant line through P and Qapproaches the tangent line to the point P on the graph of f. • and consequently, the slope mSof the secant line approaches the slope mTof the tangent line to the point P on the graph of f.

  17. Instantaneous Rate of Change of f at x= a That is, the limiting value, as h gets increasingly smaller, of the difference quotient is the slope mT of the tangent line to the graph of the functionf (x) at x = a. (slope of secant line)

  18. tends to, or approaches Instantaneous Rate of Change of f at x= a This is usually written as That is, mS approaches mT as h tends to 0.

  19. Instantaneous Rate of Change of f at x= a More briefly using symbols by writing and read as “the limit as h approaches 0 of ”

  20. Instantaneous Rate of Change of f at x= a that is, the limit symbol indicates that the value of can be made arbitrarily close to mT by taking h to be a sufficiently small number.

  21. Instantaneous Rate of Change of f at x= a Definition: The instantaneous rate of change of f (x) at x = a is defined to be the slope of the tangent line to thegraph of the function f (x) atx= a. That is, Remark: The slope mT gives a precise indication of how fast the graph of f (x) is increasing or decreasing at x = a.

  22. A Concrete Example Consider the function f (x)  – x2/4+ 9/4, whose graph is given below, at the points a – 3 and a – 1 At which of these two points is the function increasing faster? Intuition says at x – 3 because we notice the graph is steeper at the point x – 3.Why?

  23. To make this more obvious we zoom in A Concrete Example Consider the function f (x)  – x2/4+ 9/4, whose graph is given below, at the points a – 3 and a – 1 Because our brain makes no distinction between the graph of f and the tangent line to the graph at the point in question.

  24. A Concrete Example Consider the function f (x)  – x2/4+ 9/4, whose graph is given below, at the points a – 3 and a – 1 We see how the tangent line basically coincides with the graph of f near the point of contact. What is the slope of each line?

  25. A Concrete Example Consider the function f (x)  – x2/4+ 9/4, whose graph is given below, at the points a – 3 and a – 1 That is, at the points of contact, the function is increasing as fast as its corresponding tangent line.

  26. A Concrete Example Consider the function f (x)  – x2/4+ 9/4, whose graph is given below, at the points a – 3 and a – 1 We now verify the claims by explicitly computing for this function, the limit,

  27. A Concrete Example For the function f (x)  – x2/4+ 9/4 at any point awe have

  28. A Concrete Example Thus, for the function f (x)  – x2/4+ 9/4 at any point awe have Therefore, as h tends to 0, mS approaches – a/2  mT .

  29. A Concrete Example For the function f (x)  – x2/4+ 9/4 at any point awe have Thus,

  30. A Concrete Example Let us try some other values of a for the same function.

  31. We need a better notation reflecting the fact that this object is a function of a Remark The example clearly indicates that the slope mT of the tangent line is itself a function of the point a we choose.

  32. The Derivative The instantaneous rate of change mT at a is also called the derivative of f at x = a and it is denoted by f'(a). That is, f'(a) is read “f prime of a” mT = f'(a) = “slope of the tangent line to f(x) at a” = “instantaneous rate of change of f at a”

  33. The Derivative The instantaneous rate of change mT at a is also called the derivative of f at x = a and it is denoted by f'(a). That is, In our previous example, the derivative of the function f (x)  – (x2– 9)/4 at any point a is given by

  34. Rates of Change Average rate of change of f over the interval [a,a+h] is Slope of the secant line through the points (a , f (a)) and (a , f (a+h)) Instantaneous rate of change of f at x=a is Slope of the tangent line at the point (a , f (a))

  35. secant line tangent line at a Tangent Line and Secant Line

  36. Using the point-slope form of the line tangent line at a Equation of Tangent Line at x = a

  37. Terminology Finding the derivative of the function f is called differentiatingf. If f'(a) exists, then we say that f is differentiableat x = a. For some functions f , f'(a)may not exist. In this case we say that the function f is not differentiable at x = a.

  38. Quick Approximation of the Derivative Recall that f '(a) is the limiting value of the expression as we make h increasingly smaller. Therefore, we can approximate the numerical value of the derivative using small values of h. h = 0.001 often works. In this case,

  39. Quick Approximation - Example Demand:The demand for an old brand of TV is given by where p is the price per TV set, in dollars, and q is the number of TV sets that can be sold at price p . Find q(190) and estimate q'(190) . Interpret your answers.

  40. Solution

  41. a190 Geometric Interpretation At a190, q(p) decreases as fast as the tangent line does, that is, at the rate mTq'(190)2.5 TVs/$ What does this means? It means that at the price of p $190, the demand will decrease by 2.5 TV sets per dollar we increase the price. If we now set the price at $200, how many TV sets do we expect to sell?

  42. Geometric Interpretation At a190, the equation of the tangent line is y2.5( p-190 )+500 Thus, when we set the price at p$200, the line shows that we expect to sell y475 TVs The actual number we expect to sell according to the demand function is q(200)476.2 476 TVs which is very close to the prediction given by the tangent line. p200

  43. Now Recall The Example The slope mT f '(a) of the tangent line depends on the point a we choose on the graph of f .

  44. Chapter 4Techniques of Differentiation with ApplicationsSections 4.1 and 4.2

  45. Techniques of Differentiation • Derivatives of Powers, Sums and Constant asMultiples • Marginal Analysis

  46. The Derivative as a Function If f is a function, its derivative function is the function whose value at x is the derivative of f at x. That is, Notice that all we have done is substituted x for a in the definition of f'(a). This way, when we are done with the algebra, the answer will be given in terms of x.

  47. The Derivative as a Function Example: Given the function

  48. Geometric Verification At any point on the graph of f , the tangent line agrees with the graph of which already is a straight line of slope 1

  49. The Derivative as a Function Example: Given the function

  50. Geometric Verification

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