1.5 The Derivative (2.7, 2.8 & 2.9)

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1 1.5. THE DERIVATIVE (2.7, 2.8 & 2.9) The Derivative (2.7, 2.8 & 2.9) The concept we are about to de ne is not new. We simply give it a new name. Often in mathematics, when the same idea seems to play a role in various and apparently unrelated problems, instead of solving each problem, mathematicians will develop a general theory and then apply it to each problem. It is what we are about to do. In the previous section and at the beginning of the semester, we saw that the idea of rate of change (instantaneous and average) played a role in the tangent problem as well as in the velocity problem. So, we put these problems aside for now, and focus on the instantaneous rate of change as a mathematical quantity. It is this quantity we will call the derivative. We will study it extensively De nitions De nition 74 (Derivative of a function at a point) The derivative of the function y = f(x) at a point x = a, denoted by f 0 (a) or dy dx is x=a f 0 f(a + h) f(a) (a) f (x) f (a) x!a x a The function f is said to be di erentiable at a whenever the above limit exists. If f is di erentiable at every number in an open interval, then it is said to be di erentiable on that interval. The derivative of a function at a point is simply the instantaneous rate of the function at that point. Whatever meanings the instantaneous rate of change had, the derivative will also have them Interpretations of the Derivative f 0 (a) represents the slope of the tangent line to the curve y = f(x) at x = a: So, the equation of the tangent line to the curve y = f(x) at x = a becomes y f(a) = f 0 (a)(x a) f 0 (a) also represents the instantaneous rate of change of y = f(x) at x = a: In particular, if s = f(t) is the distance travelled by an object, then f 0 (a) represents the velocity of the object at time t = a. If y = f (x), then f 0 (a) measures how y is changing whenever x changes units of y by 1 unit. In fact, the units of the derivative are units of x.

2 48 CHAPTER 1. LIMITS AND CONTINUITY Example 75 Suppose that h (t) gives the height of a child (in inches) as a function of the child s age (in years). The statement h (12) = 60 means that when the child was 12 years old, the child was 60 inches tall. The statement h 0 (12) = 2 means that the child was growing at the rate of 2 in/year when the child was 12 years old. Example 76 If P (t) represents the population of the US in millions of people as a function of time (in years), then the statement P (2007) = 301 means that there were 301 million inhabitants in the US. P 0 (2007) = 3 means that in 1980, the US population was increasing at a rate of 3 million people per year. Example 77 If s (t) gives the position (in meters) of a moving object as a function of time (in seconds) then the statement s (5) = 50 means that the object has moved 50 meters when t = 5 seconds. The statement s 0 (5) = 12 means that when t = 5 seconds, the object is moving at a velocity of 12 m=s Computation of the Derivative If the function y = f (x) is given by a formula, one can use the de nition of the derivative to compute f 0 (a). If the function is given by its graph, one can estimate the slope of the tangent to nd the derivative (since f 0 (a) is the slope of the tangent). The slope can be estimated by taking two points on the graph, estimating their coordinates and using the slope formula (if the two points are (x 1 ; y 1 ) and (x 2 y 2 ) then the slope m is given by m = y 2 y 1 ). Remember, this x 2 x 1 is only an estimate. If the function is given by a table of values, one can approximate f 0 (a) by averaging the average rate of change around a. One can approximate the derivative using computer software or an advanced calculator. For example, on a TI82./83/85, the nderiv function can approximate f 0 (a) for some number a. Computers can do better. Several software packages such as scienti c Workplace, Maple, Mathematica,... can nd the derivative function, that is given f (x), they can nd the formula for f 0 (x). Example 78 Find f 0 (2) for f (x) = p x We illustrate this computation using both de nitions of the derivative, De nition 1: f 0 (a) h!0 f (a + h) f (a) h

3 1.5. THE DERIVATIVE (2.7, 2.8 & 2.9) 49 If we apply the de nition to our problem, we get: f 0 f (2 + h) f (2) (2) p p 2 + h 2 p p p p 2 + h h + 2 p 2 + h + p 2 p 2 p h 2 h!0 h p 2 + h + p 2 h!0 (2 + h) (2) h p 2 + h + p 2 h!0 h h p 2 + h + p 2 1 p p h!0 2 + h + 2 = 1 2 p 2 De nition 2: f 0 f (x) f (a) (a) x!a x a If we apply this de nition to our problem, we get: f 0 f (x) f (2) (2) x!2 x 2 p p x 2 x!2 x 2 p p p p x 2 x + 2 x!2 (x 2) p x + p 2 x!2 x 2 (x 2) p x + p 2 1 p p x!2 x = 2 p 2 Obviously, we got the same answer. Example 79 Approximate f 0 (2) if f is given by the table: x f (x) We approximate f 0 (2) by averaging the average rate of change around x = 2, that is between x = 1 and x = 2, and between x = 2 and x = 3.

4 50 CHAPTER 1. LIMITS AND CONTINUITY The average rate of change between x = 1 and x = 2 is: f (2) f (1) 2 1 = 2 ( 4) 2 1 = 2 The average rate of change between x = 2 and x = 3 is: f (3) f (2) ( 2) = 3 2 = 14 The average of these two values is 8, so, f 0 (2) t 8. Example 80 Find f 0 (1) if f is given by gure 1.21 (note: The graph below shows both f and its tangent at x = 1. On a test problem, you would have to draw the tangent). Figure 1.21: Function and its tangent f 0 (1) is the slope of the tangent at x = 1. The tangent is already drawn. We estimate the slope of the tangent by picking two points on it. For example, we see that the tangent goes through the origin, so (0; 0) is on the tangent. Also, (1; 2) is on the tangent. So, it slope is = 2. Thus, f 0 (1) t 2.

5 1.5. THE DERIVATIVE (2.7, 2.8 & 2.9) Some Results about the Derivative The same way we can compute the derivative of a function f at a point x = a, we can also compute the derivative function. Its de nition is similar to that of the derivative at a point, except that it applies to any point. De nition 81 (derivative function) The derivative of a function f denoted f 0 (x) or dy (assuming we write y = f (x)) is dx f 0 (x) h!0 f (x + h) f (x) h The answer will be a function of x. To compute this derivative function, you can also use both formulas. If you use f 0 f (x + h) f (x) (x), then you will get the answer, a function of x. If you use f 0 f (x) f (a) (a), you x!a x a will get an answer in terms of a. Since the answer is true for every a, you can replace a by x. We illustrate this with examples. Example 82 Find f 0 (x) for f (x) = x 2. Using f 0 f (x + h) f (x) (x), we get: f 0 (x) = f (x + h) f (x) lim = (x + h) 2 x 2 lim = x 2 + 2xh + h 2 x 2 lim = 2xh + h 2 lim = h (2x + h) lim (2x + h) h!0 = 2x So, if f (x) = x 2 then f 0 (x) = 2x. Knowing such a formula means that if we need to evaluate f 0 (x) for various values of x, we can use that formula. For example, f 0 (2) = 2 (2) = 4. f 0 (10) = 2 (10) = 20. f 0 (0) = 2 (0) = 0. Example 83 Find f 0 (x) for f (x) = p x.

6 52 CHAPTER 1. LIMITS AND CONTINUITY Using f 0 f (x) (a) x!a x f (a), we get a f 0 f (x) f (a) (a) x!a x a p p x a x!a x a ( p p p p x a) ( x + a) x!a (x a) ( p x + p a) x a x!a (x a) ( p x + p a) 1 p p x!a x + a 1 = 2 p a Since this is true for every a, we see that if f (x) = p x, then f 0 (x) = 1 2 p x. We see in particular that f 0 (2) = 1 2 p 2 What does it mean for f to be di erentiable? Theorem 84 If a function f is di erentiable at a point x = a, then f is continuous at x = a. Proof. We assume that f is di erentiable at x = a and we must prove f is continuous at x = a, that is lim f (x) = f (a). Since f is di erentiable at x = a, x!a f 0 (a) exists and f 0 f (x) f (a) (a). We need to connect this to what we x!a x a want to prove. We start with f (x) f (a) and multiply it by x a. so, we get x a f (x) f (a) = f (x) f (a) x a Then, we take the limit as x! a, we get f (x) lim [f (x) f (a)] x!a x!a x f (x) x!a x (x a) f (a) a f (a) a (x a) h lim x!a (x This is the product rule of limits. It applies since both limits exist. The rst one exists because by assumption, f is di erentiable at x = a. So, we get h i lim [f (x) f (a)] = f 0 (a) lim (x a) x!a x!a = f 0 (a) [0] = 0 a) i

7 1.5. THE DERIVATIVE (2.7, 2.8 & 2.9) 53 but lim [f (x) x!a that lim f (x) x!a f (a)] f (x) lim f (a) f (x) x!a x!a x!a f (a) = 0 or lim f (x) = f (a). x!a f (a). So, we see This means that being di erentiable is a stronger condition than being continuous. It also implies that if a function is not continuous at a point, then it cannot be di erentiable there. There are also other points where a function might not be di erentiable. A function f is not di erentiable at x = a if one of the conditions below is satis ed: 1. f is not continuous at x = a 2. The graph of f has a corner point at x = a. This is the case for example of f (x) = jxj at x = The graph of f has a vertical tangent at x = a. Higher order Derivatives De nition 85 (second order derivative) The second order derivative of a function f, denoted f 00 (x) or d2 y dx 2 is f 00 (x) = (f 0 (x)) 0 In other words, it is the derivative of f 0. Remark 86 If f (x) gives the position of an object, we already know that f 0 (x) gives the velocity of the object. f 00 (x) gives the acceleration of the object. Similarly, we can de ne the third order derivative as being the derivative of the second derivative, and so on. The notation for these derivatives is: Third order: f 000 (x) or d3 y dx 3 is the derivative of f 00 (x). De nition 87 Fourth order: f (4) (x) or d4 y dx 4 is the derivative of f 000 (x). nth order: f (n) (x) or dn y dx n Note the notation change for derivative or order 4 or higher.

8 54 CHAPTER 1. LIMITS AND CONTINUITY Meaning of the sign of f 0 and f 00 First, we look at the meaning of the sign of f 0. We know that f 0 (a) gives the slope of the tangent to y = f (x) at x = a. If f is increasing at x = a, its tangent will be an increasing line, hence will have a positive slope. Similarly, if f is decreasing at x = a, then its tangent will be a decreasing line hence will have a negative slope. So, we have the following result, which we ll be able to prove later on in the semester. Proposition 88 The sign of f 0 decreasing as follows: is related to a function being increasing or If f 0 (x) > 0 on an interval then f is increasing on that interval. If f 0 (x) < 0 on an interval then f is decreasing on that interval. We can use the graph of f to get information about the sign of f 0. We can also use the graph of f 0 to know where f is increasing and decreasing. The examples below illustrate these techniques. Example 89 The graph of f (x) is given in gure Answer the questions below. 1. On which intervals is f (x) increasing, decreasing? From the graph, we can see that f is decreasing on ( on (0; 1). 1; 0) and increasing 2. On which intervals is f 0 (x) positive, negative, zero? Since f 0 (x) > 0 where f is increasing, we see that f 0 (x) > 0 on (0; 1). Similarly, f 0 (x) < 0 on ( 1; 0). Example 90 The graph of f 0 (x) is given in gure Answer the questions below. 1. On which intervals is f increasing, decreasing? Since f is increasing where f 0 (x) > 0, we see that f is increasing on (1; 1). Similarly, it is decreasing on ( 1; 1). 2. For which value of x does f seem to have a maximum or a minimum? f appears to have a minimum when x = 1 since it is decreasing to the left of x = 1 and increasing to the right. Above, we saw that the sign of f 0 could give us information about where f is increasing and decreasing. However, a function can be increasing (or decreasing) di erent ways. Figure 1.24 shows a function increasing at a constant rate, one at an increasing rate, one at a decreasing rate. In the three cases, the sign of the rst derivative would be positive. So, though the rst derivative allows us to determine if and where a function is increasing, it does not really tells us how a function is increasing.

9 1.5. THE DERIVATIVE (2.7, 2.8 & 2.9) 55 Figure 1.22: Graph of y = f (x) Figure 1.23: Graph of y = f 0 (x)

10 56 CHAPTER 1. LIMITS AND CONTINUITY Figure 1.24: Three increasing functions For this, we need to look at the second derivative. It works as follows. In the case of the blue function in gure 1.24, we see that not only the function is increasing, but so will the slope of its tangent. In other words, f 0 (x) would also be increasing. But if f 0 (x) is increasing, then its derivative, f 00 (x) must be positive. Similarly, we can see that if f is decreasing at a slower and slower rate, then f 00 (x) must be negative. So, to see whether a function is increasing, we look at the sign of f 0. To see how a function is increasing, we look at the sign of f 00. We begin with preliminary de nitions. De nition 91 (Concave up, down) The graph of a function is said to be concave up on an interval if it is above its tangents in that interval. It is concave down if it is below its tangents in that interval. De nition 92 (In ection point) An in ection point of a function f is a point where the concavity of the graph of f changes. From what we stated above, we have the following proposition. Proposition 93 The sign of f 00 is related to concavity as follows: If f 00 (x) > 0 on an interval then f is concave up on that interval. If f 00 (x) < 0 on an interval then f is concave down on that interval. We can use the graph of f to get information about the sign of f 00. We can also use the graph of f 00 to know where f is concave up or down. The examples below illustrate these techniques.

11 1.5. THE DERIVATIVE (2.7, 2.8 & 2.9) 57 Figure 1.25: Graph of y = f (x) Example 94 The graph of f (x) is given in gure Answer the questions below. 1. On which intervals is f concave up, down? The graph of f seems to be concave down on ( (0; 1). 1; 0) and concave up on 2. On which intervals is f 00 (x) positive, negative? Since f 00 (x) > 0 when f is concave up, it follows that f 00 (x) > 0 on (0; 1) and f 00 (x) < 0 on ( 1; 0). 3. Does the graph of f seem to have an in ection point? If yes, for which value of x? An in ection point is a point where the concavity changes. This happens at x = 0. Example 95 The graph of f 00 (x) is given in gure Answer the questions below? 1. Where is f concave up, down? f is concave down where f 00 (x) < 0 that is on ( where f 00 (x) > 0 that is on (1; 1). 1; 1). It is concave up 2. Is there an in ection point, where? An in ection point is a point where the concavity changes. This happens at x = 1.

12 58 CHAPTER 1. LIMITS AND CONTINUITY Figure 1.26: Graph of y = f 00 (x) Sample problems: 1. When an object is thrown upward, its altitude h is given as a function of time by the following relation: h = 16t t + 6: In this relation, t is expressed in seconds, h in feet. (a) Find the average velocity of the object over the time intervals [0; 1], [0; 2], [1; 2] (b) Find the instantaneous velocity of the ball after 1 second, after 2 seconds, after 3 seconds. (c) What will be the maximum altitude reached by the object? (d) When will the object reach the ground? 2. Find the equation of the tangent line to the curve y = x when x = 0, when x = 1: 3. The population (in thousands) of San Jose, California is given by the table Year (t) P (t) (a) Find the average rate of growth of the population from 1986 to Give units to your answer. (b) Find the average rate of growth of the population from 1988 to Give units to your answer.

13 1.5. THE DERIVATIVE (2.7, 2.8 & 2.9) 59 (c) Express the instantaneous rate of growth of the population in 1992 as a mathematical quantity, then use the table to estimate it. Give units to your answers. 4. Let h (t) represent the altitude of a plane (in feet) t seconds after the plane has taken o. Express as a mathematical quantity the rate at which the plane is climbing 5 minutes after take o. Give units to your answers. 5. Looking at problem 3 on page 145, assuming the function whose graph is given is called f (x), answer the following questions: (a) What is the sign of f (x) at A, B, C, D, E. (b) Same question for the sign of f 0. (c) What does the sign of f 0 seem to suggest about the function f? 6. Find f 0 (1), f 0 (2) for f (x) = 1 x. 7. Let f (x) represent the elevation of the Mississippi river x miles from its source. What is the sign of f 0 (x)? 8. Let P (t) represent the population of the United States as a function of time (in years), Let Q (t) be the population of Mexico. (a) Which of these two quantities is larger: P (1990) or Q (1990)? why? (b) Which of these two quantities is larger: P 0 (1990) or Q 0 (1990)? why? 9. Let D (t) represent the value of the stock market (the Dow Jones Industrial Index). What do you think the sign of D 0 (t) has been these past few months? why? 10. Sketch the graph of a function f satisfying all the conditions below (a) f 0 (x) < 0 when x < 1 or x > 4 (b) f 0 (x) > 0 when 1 < x < 4 (c) f 0 ( 1) does not exist, f 0 (4) = 0 (d) f is continuous for all reals. 11. Sketch the graph of a function f satisfying all the conditions below (a) f 0 (x) > 0 when x < 1 or x > 4 (b) f 0 (x) < 0 when 1 < x < 4 (c) f 0 ( 1) = f 0 (4) = 0 (d) f is continuous for all reals. 12. Same question as above, add the condition that f 00 (x) < 0 when x < 2 and f 00 (x) > 0 when x > 2

14 60 CHAPTER 1. LIMITS AND CONTINUITY 13. Same question as above, replace the condition on f 00 by f 00 (x) < 0 when x < 2 or x > 6 and f 00 (x) > 0 when 2 < x < In your book, do # 1, 3, 4, 5, 7, 13, 18, 19, 21, 23, 25, 29 on page In your book, do # 1, 3, 19, 23, 31, 32, 33, 34, 47 on pages