2.8 The derivative as a function

Transcription

1 CHAPTER 2. LIMITS Te derivative as a function Definition. Te derivative of f(x) istefunction f (x) defined as follows f f(x + ) f(x) (x). 0 Note: tis differs from te definition in section 2.7 in tat we don t ave te prase at te point x = a. As a result, in tis definition f (x) isafunction, not a number. Comments. We interpret f (x) as giving a formula for te slope of te tangent line, te velocity, etc. Example. Find te derivative of f(x) = and te equation of te tangent line at x = 4. Solution. f f(x + ) f(x) (x) 0 + x x (x + ) x ( + + ) ( + + ) 0 = = ( + + ) = 2 x + + x + + Now, to find te equation of te tangent line we fill te following in: y = m(x 4) + y 0 m = f (4) y 0 = f(4) = 4=2 Te main difference between tis example and te ones in te previous section is ow we find f (4). Before, we set up a limit tat ad 4 in it to find f (4). Tis time, we calculated f (x) as a function, and we ll plug 4 into tis function m = f (4) = 2 4 = 4

2 CHAPTER 2. LIMITS 57 Putting it all togeter we ave and te grap looks like tis: y = (x 4) Example 2. Find f (x) for f(x) = x 2. Solution. f (x) 0 0 (x+) 2 x 2 x 2 x 2 (x+) 2 (x+)2 x 2 (x+) 2 x 2 (x + ) 2 0 x 2 (x + ) 2 x 2 (x 2 +2x + 2 ) 0 x 2 (x + ) 2 2x 2 0 x 2 (x + ) 2 0 ( 2x ) x 2 (x + ) 2 2x 0 x 2 (x + ) 2 = 2x 0 x 2 (x + 0) 2 = 2x x 4 = 2 x 3 Notation for derivative: Given a function f(x) tere are oter ways to write te derivative: f df (x) or dx.

3 CHAPTER 2. LIMITS 58 Te notation df is called Leibniz notation (because it is named after Gottfried dx Leibniz). If y is a function of x we ave variations on tis: y or We also use just part of te Leibniz notation: d. Tis stands for te prase dx take te derivative (wit respect to x) of.... Tere are some advantages of Leibniz notation: You don t ave to name f(x), Tis notation suggests a rate of cange, suc as y x. tis notation makes explicit te role of x in taking te derivative, tis notation makes te cain rule (coming later) look nice, tis notation reminds us tat d is an operator (i.e. it is a ting tat we apply dx to functions), tis notation is easier to see tan. dy dx. Example 3. Write te answer to Example and Example 2 in Leibniz notation. Solution. Example becomes and Example 2 becomes d x = dx 2 d dx x 2 = 2 x 3. At tis point we turn to estimating derivatives grapically. Example 4. Te grap below defines a function f(x) Use te given grap to produce a roug sketc of te grap of te derivative f (x).

4 CHAPTER 2. LIMITS 59 Solution. Let s look at te given grap and identify were te slope is 0: tis is were te grap flattens out and becomes orizontal for an instant. Tere are tree points were tis appens: x = A, x = B and x = C. Now, we turn slopes into y-values: slope of f is 0 at x = A, B, C y-value of f is 0 at x = A, B, C We fill in tis part of te grap: Furtermore, we know tat between A and B, f is going up, and to te rigt of C, f is going up. We turn tese slopes into y-values: slope of f is + between x = A and x = B, and to te rigt of C y-value of f is + between x = A and x = B, and to te rigt of C

5 CHAPTER 2. LIMITS 60 Finally, we know wen f sould be negative: between x = 0 and x = A, f is going down, and between x = B and x = C, f is going down. We turn tese slopes into y-values: slope of f is between x =0 and x = A, and between x = B and x = C y-value of f is between x = 0 and x = A and between x = B and x = C Note: te grap we ave above is made up, altoug it is made up to ave te rigt general sape. For curiosity, ere is te real grap of f (x). It was produced using knowledge tat I kept idden in tis example, te real formula for f(x).

6 CHAPTER 2. LIMITS 6 Tis is were we ended on Friday, October 4 Given a function f(x), te derivative f (x) is also a function. Since f (x) isa function we can take te derivative again, and get f (x) te second derivative. We can (but rarely will) take te derivative again and again: f(x) = original function f (x) = te derivative of f(x) f (x) = te derivative of f (x): te second derivative f (x) = te derivative of f (x): te tird derivative f (4) (x) = te derivative of f (x): te fourt derivative etc. Example 5. Te grap below sows tree graps a, b and c. One of tese graps equals a function f, one equals f and one equals f. Identify wic grap is wic. Solution. Probably we need to work wit just one function at a time. Start wit a. Were is it orizontal? It looks like a is orizontal near x =0.2, x = 3 and

7 CHAPTER 2. LIMITS 62 x =6.5. In oter words, Now we turn tese slopes into y-values: slope of a is 0 near x =0.2, 3, 6.5. Wic grap as y-values of 0 near x =0.2, 3, 6.5? Te answer is b. In oter words, a = b. Now we do it again. Let s look at b. Were is it orizontal? It looks like b is orizontal near x =.5 and x =4.5. In oter words, Now we turn tese slopes into y-values: slope of b is 0 near x =.5, 4.5. Wic grap as y-values of 0 near x =.5, 4.5? Te answer is: none of tem. In oter words, b is not sown. Let s do it for te last grap, c. Were is it orizontal? It looks like c is orizontal near x = 2 and x = 4. In oter words, Now we turn tese slopes into y-values: slope of c is 0 near x =2, 4. Wic grap as y-values of 0 near x =2, 4? Te answer is: a. In oter words, c = a. So now we ave c = a and a = b. Tat means a =(c ) = c. In oter words, f = c, f = a and f = b.