2.2 The derivative as a Function

Transcription

1 2.2 The derivative as a Function

2 Recall: The derivative of a function f at a fixed number a: f a f a+h f(a) = lim h 0 h Definition (Derivative of f) For any number x, the derivative of f is f x f x+h f(x) = lim h 0 h

3 Remark f is a new function derived from f. So, it is called derivative. Domain of f = x f x exists} f (x) can be interpreted geometrically as the slope of the tangent line to the graph of f at the point x, f x if it exists.

4 Example The graph of a function f is given. Use it to sketch the graph of the derivative f.

5 Example(Answer) The graph of a function f is given. Use it to sketch the graph of the derivative f.

6 Example The graph of a function f is given. Use it to sketch the graph of the derivative f.

7 Example(Answer) The graph of a function f is given. Use it to sketch the graph of the derivative f.

8 Example a. If f x = x 3 x, find a formula for f (x). b. Illustrate by comparing the graphs of f and f.

9 Example(Hint) a. f f x+h f(x) x = lim x 0 h b. = 3x 2 1

10 Example Let f x = x. a. Find the derivative of f. b. State the domain of f.

11 Example(Hint)

12 Example Find f if f x = 1 x 2+x. Answer: 3 (2 + x) 2

13 Notations f x Df x = y = dy = lim dx = D x f x x 0 y x = d dx f x = D, d : differential operators ( meaning dx operation of differentiation )

14 Notation dy dx : Leibniz notation Differentiation at x = a: f a = dy dx x=a = dy dx ] x=a

15 Definition A function f is differentiable at a if f a exists. f is differentiable on an open interval (a, b) [or (a, ), (,a) or (, )] if f is differentiable at every number in the interval.

16 Example Where is the function f x = x differentiable? Hint: Divide the cases into when x > 0 and x < 0.

17 Theorem If f is differentiable at a, then f is continuous at a. Remark: The converse of the above theorem is not true.

18 How can a function fail to be differentiable? Vertical tangent line (lim x c f x = ) Corner

19 How can a function fail to be Discontinuity differentiable?

20 Higher Derivatives f = f = f = d dx dy dx f = d dx ( d dx dy dx ) f (n) = dn y dx n

21 Example If f x = x 3 x, find f x, f x and f n x. Remark: All higher order derivatives whose order is greater than the order of the given polynomial vanishes.

22 2.3. Differentiation Formulas

23 Derivative of a constant function Hint: d dx Verify straightforwardly. c = 0, c: number Verify geometrically by relating the definition with the graph.

24 Derivative of Power Functions d (x) = 1 dx d dx x2 = 2x d dx xn = nx n 1

25 The Constant Multiple Rule If c is a constant and f is a differentiable function, then d cf x dx = c d dx f(x)

26 Example Differentiate 1. f x = x 6 2. f t = t 4 3. g x = 3x 4 4. h x = x 2

27 The Sum and difference Rule If f and g are both differentiable, then d dx f x + g x = d dx f x + d dx g(x) d dx f x g x = d dx f x d dx g(x)

28 Example d dx (x8 + 12x 5 4x x 3 6x + 5)

29 Example Find the points on the curve y = x 4 6x 2 + 4, where the tangent line is horizontal. Answer: (0,4), ( 3, 5), ( 3, 5)

30 Example The equation of motion of a particle is s = 2t 3 5t 2 + 3t + 4, where s is measured in centimeters and t in seconds. Find the acceleration as a function of time. What is the acceleration after 2 seconds?

31 The Product Rule If f and g are both differentiable, then d f x g x dx = f x d dx g x + g x d dx [f x ] The derivative of a product of two functions is the first function times the derivative of the second function plus the second function times the derivative of the first function.

32 Example Differentiate f x = (6x 3 )(7x 4 )

33 Example If h x = xg x, and it is known that g 3 = 5 and g 3 = 2, find h 3.

34 The Quotient Rule If f and g are differentiable, then f g = gf fg The derivative of a quotient is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator. g 2

35 Example Let y = x2 +x 2 x Find the derivative y.

36 Power Rule for negative power function If n is a positive integer, then d dx x n = nx n 1

37 Example Differentiate 1. y = 1 x 2. y = 6 t 3

38 The Power Rule( General Version ) If n is any real number, then d dx (xn ) = nx n 1

39 Example Differentiate 1. f x = x π 2. y = 1 3 x 2 3. f t = t (a + bt)

40 Example Find equations of the tangent line and normal line to the curve y = x/(1 + x 2 ) at the point 1, 1. 2 Answer: Tangent: y 1 2 = 1 4 x 1, y = 1 4 x Normal: y 1 2 = 4 x 1, y = 4x 7 2

41 Example At what points on the hyperbola xy = 12 is the tangent line parallel to the line 3x + y = 0? Answer: (2,6), ( 2, 6)

42 2.4. Derivatives of Trigonometric Functions

43 Properties of Trigonometric function Trigonometric functions, such as sine, cosine, tangent, cotangent, cosecant and secant functions are defined for all real numbers x. For example, f (x) = sin x. Here, real numbers are radian measure, rather than degree. All of the trigonometric functions are continuous at every number in their domains.

44 Derivative of sine function is cosine function. (Graphical and intuitive explanation)

45 Two important formula lim θ 0 sin θ θ = 1 cos θ 1 lim θ 0 θ = 0

46 Example Find lim x 0 sin 7x 4x. Hint: Use 1. sin 7x 4x = 7 4 sin 7x 7x 2. lim θ 0 sin θ θ = 1

47 Example Calculate lim x 0 x cot x. Hint: Use 1. cot x = cos x sin x 2. lim θ 0 sin θ θ = 1

48 Derivatives of Trigonometric Functions d dx (sin x) = cos x d dx (csc x) = csc x cot x d dx (cos x) = sin x d dx (sec x) = sec x tan x d dx (tan x) = sec2 x d dx (cot x) = csc2 x

49 Example Differentiate y = x 2 sin x.

50 Use of Trigonometric Functions Trigonometric functions are often used in modeling real-world phenomena. In particular, vibrations, waves, elastic motions, and other quantities that vary in a periodic manner can be described using trigonometric functions. In the following example we discuss an instance of simple harmonic motion.

51 Example An object at the end of a vertical spring is stretched 4 cm beyond its rest position and released at time t = 0. (See Figure and note that the downward direction is positive.) Its position at time t is s = f (t) = 4 cos t Find the velocity and acceleration at time t and use them to analyze the motion of the object.

52 Example( Hint ) The velocity and acceleration are v = ds = dt 4 sin t and a = dv = 4 cos t. dt The object oscillates from the lowest point (s = 4 cm) to the highest point (s = 4 cm). The period of the oscillation is 2, the period of cos t.

53 Example ( Further ) The speed is v = 4 sin t, which is greatest when sin t = 1, that is, when cos t = 0. So the object moves fastest as it passes through its equilibrium position (s = 0). Its speed is 0 when sin t = 0, that is, at the high and low points. The acceleration a = 4 cos t = 0 when s = 0. It has greatest magnitude at the high and low points. See the graphs in the Figure.

54 2.5. The Chain Rule

55 How can we differentiate F x = x 2 + 1? We know the derivative formula for y = y = x We may view F(x) as a composite function: x and Then, Use the Chain rule.

56 Chain Rule

57 Example Find F x if F x = x Hint: 1. Analyze the given function to write that as a composite function. 2. Let the inner function x as u. 3. Write the outer function as a function of u. That is, y = u. 4. Apply the Chain rule: dy 5. Substitute x for u. = dy dx du du dx.

58 Chain Rule for Sine and Cosine Functions If y = sin u, where u is a differentiable function of x, then dy dx = (cos u) du dx. If y = cos u, where u is a differentiable function of x, then dy dx = ( sin u) du dx.

59 Example Differentiate 1. f x = sin x 4 + 2x g x = cos(x 2 + 1)

60 The Chain Rule for the Power Function This rule is applied to the functions such as (3x + 1) 56, (5x 7 3x ) 11 etc.

61 Example Differentiate 1. f x = (3x + 1) g x = (5x 7 3x ) 11

62 2.6. Implicit Differentiation

63 How can we differentiate a smooth curve such as x 2 + y 2 = 25, x 3 + y 3 = 6xy? The collection of the points which satisfies the equation on the Cartesian plane is not the graph of a function because that doesn t pass the vertical line test. However, we still can consider the concept of slope of the graph locally at most points of the graph. Implicit differentiation makes the calculation possible.

64 Example If x 2 + y 2 = 25, find dy dx. Hint: 1. Differentiate as usual if the term is associated with only x. 2. Use the chain rule when the term contains y.

65 Example Find an equation of the tangent to the circle x 2 + y 2 = 25 at the point (3, 4). Answer: 3x + 4y = 25

66 a. Find y if x 3 + y 3 = 6xy. b. Find the tangent to x 3 +y 3 = 6xy at the point (3,3). c. At what point in the first quadrant is the tangent line horizontal? Example

67 Answer a. y = 2y x2 y 2 2x b. y 3 = 1 x 3 c. y = 0 when 2y x 2 = 0 if y 2 2x 0. (0,0) is a double point which has two tangent lines. The tangent is horizontal at (2 4 3, 2 5 3). (Remark: 4 3 and 5 3 are exponents.)

68 Example Find y if x 4 + y 4 = 16. Answer: y = x3 y 3 y = 48x2 y 7

69 Linear Approximation and Differentials

70 Notion of derivative Local figure of a curve

71 By zooming in toward a point on the graph of a differentiable function, the graph looks more and more like its tangent line. That is, f (x) f (a) + f (a)(x a) near the point (a, f(a)).

72 Definitions f (x) f (a) + f (a)(x a) is called the linear approximation or tangent line approximation of f at a. The linear function whose graph is this tangent line, that is, L (x) = f (a) + f (a)(x a) is called the linearization of f at a.

73 Example Find the linearization of the function f (x) = x + 3 at x = 1 and use it to approximate the numbers 3.98 and Are these approximations overestimates or underestimates? Hint:

74 Example Answer: Overestimate

75 Linear Approximations and Differentials The tangent line approximation gives good estimates when x is close to 1 but the accuracy of the approximation deteriorates when x is farther away from 1.

76 Numerical Evidence

77 Differentials If y = f (x), where f is a differentiable function, then the differential dx is an independent variable; that is, dx can be given the value of any real number. The differential dy is then defined in terms of dx by the equation dy = f (x) dx Thus, dy is a dependent variable; it depends on the values of x and dx.

78 The geometric meaning of differentials

79 Example Compare the values of y and dy if y = f (x) = x 3 + x 2 2x + 1 and x changes (a) from 2 to 2.05 and (b) from 2 to Hint and Answer: (a) f(2)=9, f(2.05)= , y= dy= (3x 2 + 2x 2) dx, dy=0.7. (b) f (2.01) = , y= , dy = 0.14