3.2 Basic Differentiation Rules and Rates of Change

Transcription

1 3.2 Basic Differentiation Rules and Rates of Change Objectives Find the derivative of a function using the Constant Rule. Find the derivative of a function using the Power Rule. Find the derivative of a function using the Constant Multiple Rule. Find the derivative of a function using the Sum and Difference Rules. Objectives Find the derivatives of the sine, cosine, and exponential functions. Use derivatives to find rates of change. 1

2 We have used the limit definition to find derivatives. In this section you will be introduced to several differentiation rules that allow you to find derivatives without the direct use of the limit definition. Notice that the Constant Rule is equivalent to saying that the slope of a horizontal line is 0. This demonstrates the relationship between slope and derivative. Figure 3.14 Let f(x) = 5. Then, by the limit definition of the derivative, f(x + x) f(x) x x x ' f ( x) = lim = lim = = 0 x 0 x 0 2

3 The Power Rule n If n is a real number, then the function f (x) = x is differentiable and d x n n-1 = n x. dx Using the Power Rule Function Derivative a. b. c. The Power Rule In Example (c), note that before differentiating, 1/x 2 was rewritten as x 2. Rewriting is the first step in many differentiation problems. 3

4 Find the derivative of the given function. f(x) = -9 f(x) = 0 f(x) = x y = y = x 5 x The Constant Multiple Rule Informally, the Constant Multiple Rule states that constants can be factored out of the differentiation process, even if the constants appear in the denominator. 4

5 The Constant Multiple Rule The Constant Multiple Rule and the Power Rule can be combined into one rule. The combination rule is The Sum and Difference Rules The Sum and Difference Rules can be extended to any finite number of functions. For instance, if F(x) = f(x) + g(x) h(x), then F (x) = f (x) + g (x) h (x). 5

6 Using the Sum and Difference Rules Function Derivative a. f(x) = x 3 4x + 5 f (x) = 3x 2 4 b. g(x) = + 3x 3 2x g (x) = 2x 3 + 9x 2 2 Derivatives of Sine and Cosine Functions We have studied the following limits. These two limits can be used to prove differentiation rules for the sine and cosine functions. 6

7 Example 8 Derivatives Involving Sines and Cosines Function Derivative a. y = 2 sin x y = 2 cos x b. c. y = x + cos x y = 1 sin x Derivatives of Exponential Functions One of the most intriguing (and useful) characteristics of the natural exponential function is that it is its own derivative. Consider the following. Let f(x) = e x. 7

8 Derivatives of Exponential Functions The definition of e tells you that for small values of x, you have e (1 + x) 1/ x, which implies that e x 1 + x. Replacing e x by this approximation produces the following. Derivatives of Exponential Functions This result is stated in the next theorem. 8

9 Derivatives of Exponential Functions Find the derivative of each function. a. f(x) = 3e x b. f(x) = x 2 + e x c. f(x) = sin x e x Solution: a. b. c. Derivatives of Exponential Functions The derivative of e x = e x. Find the derivative of each function. y = x 3 + 2e x 3 x y = e + 2cos x 4 3 x 6 f ( x) = 2 x 2 y = 3x(6x 5x 2 ) f ( x) = + 5cos x 3 x y = 4 3 x 9

10 Finding an Equation of a Tangent Line 1.Find the point on the graph of f at the given value. 2.Find the slope of the graph at the point. 3.Use the point-slope form of the equation of a line to find the equation of the tangent line. Find an equation of the tangent line to the graph of f at the given point. y = x 4 x at (-1, 2) f(x) = sinx + 1/2e x at(π, ½ e π ) ' 1 f ( x) = cos x + e 2 To find the slope of the line tangent at the given point, substitute π for each x in the derivative. 1 π 1 m = cos π + e = 1+ e 2 2 π To find the equation of the tangent line use y - y1 = m(x - x 1). 1 π 1 π y e = ( 1 + e )( x π ) y = 1+ e x + e + π πe π π π x 10

11 Determine the point(s) (if any) at which the graph of the function has a horizontal tangent line. What is the slope of a horizontal line? y = x 3 + x ' 2 y = 3x x So, there are no horizontal tangent lines to the graph of this function. y = 3x + 2cos x, 0 x < 2π ' y = 3 2sin x 3 2sin x = 0 2sin x = 3 sin x = 3 2 π 2π x =, x = 3 3 Find the y-values for each of the x-values. π π y = 3 + 2cos 3 3 3π 1 y = π y = y = 3π π 3π + 3 2π 2 3π 3 Answers :, ;, Rates of Change We have seen how the derivative is used to determine slope. The derivative can also be used to determine the rate of change of one variable with respect to another. Applications involving rates of change occur in a wide variety of fields. A few examples are population growth rates, production rates, water flow rates, velocity, and acceleration. A common use for rate of change is to describe the motion of an object moving in a straight line. 11

12 Rates of Change In such problems, it is customary to use either a horizontal or a vertical line with a designated origin to represent the line of motion. On such lines, movement to the right (or upward) is considered to be in the positive direction, and movement to the left (or downward) is considered to be in the negative direction. The function s that gives the position (relative to the origin) of an object as a function of time t is called a position function. Rates of Change If, over a period of time t, the object changes its position by the amount s = s(t + t) s(t), then, by the familiar formula the average velocity is Average velocity 12

13 Finding Average Velocity of a Falling Object If a billiard ball is dropped from a height of 100 feet, its height s at time t is given by the position function s = 16t Position function where s is measured in feet and t is measured in seconds. Find the average velocity over each of the following time intervals. a. [1, 2] b. [1, 1.5] c. [1, 1.1] Solution For the interval [1, 2], the object falls from a height of s(1) = 16(1) = 84 feet to a height of s(2) = 16(2) = 36 feet. The average velocity is = 48 feet per second. 13

14 Rates of Change Suppose that in last example you wanted to find the instantaneous velocity (or simply the velocity) of the object when t = 1. Just as you can approximate the slope of the tangent line by calculating the slope of the secant line, you can approximate the velocity at t = 1 by calculating the average velocity over a small interval [1, 1 + t] (see Figure 3.21). The average velocity between t 1 and t 2 is the slope of the secant line, and the instantaneous velocity at t 1 is the slope of the tangent line. Figure 3.21 Rates of Change By taking the limit as t approaches zero, you obtain the velocity when t = 1. Try doing this you will find that the velocity when t = 1 is 32 feet per second. In general, if s = s(t) is the position function for an object moving along a straight line, the velocity of the object at time t is Velocity function In other words, the velocity function is the derivative of the position function. 14

15 Rates of Change Velocity can be negative, zero, or positive. The speed of an object is the absolute value of its velocity. Speed cannot be negative. The position of a free-falling object (neglecting air resistance) under the influence of gravity can be represented by the equation Position function where s 0 is the initial height of the object, v 0 is the initial velocity of the object, and g is the acceleration due to gravity. Rates of Change On Earth, the value of g is approximately 32 feet per second per second or 9.8 meters per second per second. 15

16 Use the position function s(t) = -16t 2 + v 0 t + s 0 for free-falling objects. A ball is thrown straight down from the top of a 220-foot building with an initial velocity of 22 feet per second. What is its velocity after 3 seconds? What is its velocity after falling 108 feet? Use the position function s(t) = -4.9t 2 + v 0 t + s 0 for free falling objects. To estimate the height of a building, a stone is dropped from the top of the building into a pool of water at ground level. How high is the building if the splash is seen 5.6 seconds after the stone is dropped? 16

17 The area of a square with sides of length s is given by A= s 2. Find the rate of change of the area with respect to swhen s= 6 meters. 17