Rules for Derivatives

Transcription

1 Chapter 3 Rules for Derivatives 3.1 Shortcuts for powers of x, constants, sums, an ifferences Leibniz notation for erivative: f 0 (x) = f, V 0 (t) = V t, C0 (q) = C q, etc. reas as the erivative of.... E.g. erivative of x 2 equals 2x. Notation for plugging in a number: x2 = 2x reas as the f 0 (5) = f x=5, V 0 (15) = V t t=15, C 0 (1200) = C q q=1200, etc. Comments. Recall that Leibniz notation, such as f reas as the erivative of f with respect to x, or take the erivative of f with respect to x. There is a useful variation of this: we let stan for the phrase the erivative of. Using this notation, we state our first bunch of erivative rules. Constant rule: C =0whereC is a constant. Linear rule: (mx + b) =m. Constant multiple rule: (C f(x)) = C f 0 (x) wherec is a constant. Sum an Di erence rule: [f(x) ± g(x)] = f 0 (x) ± g 0 (x) Power rule: xn = nx n 1 where n is any real number. This is where we ene on Monay, February 18 48

2 CHAPTER 3. RULES FOR DERIVATIVES 49 Example 1. Fin the erivative of y =3.7x 5 253x x 2 + 7; use at most one of the above rules at a time, an inicate which rule this is. (3.7x5 253x x 2 + 7) = (3.7x 5 ) 0 (253x 4 ) 0 + (10x 2 ) (sum an i erence rule) =3.7(x 5 ) 0 253(x 4 ) (x 2 ) (Constant multiple rule an constant rule) =3.7 5x x x 1 (power rule) = 18.5x x x (cleaning up) Example 2. We return to the problem pose in Section 2.1, Example 1 an Section 2.2, Example 1. Recall that the ball ha a position given by p(t) = 4.9t t + 2. Fin a formula for the velocity of the ball at time t. vel = eriv of position v(t) =p 0 (t) = t ( 4.9t t + 2) = 4.9(2)t (1)t 0 +0 = 9.8t Example 3. (a) Let f(x) =7x 3 7 x Fin f 0 (x). (b) Let C(q) = 165 p q be a total cost function. Fin the marginal cost at q = 5. (a) First we rewrite f(x) in power-of-x-form f(x) =7x 3 7x Don t be confuse, sometimes people make the change from 7 x 3 to 7x 3 an think that s the erivative. No! That s just the efinition of a negative power. The erivative comes next f(x) =7x 3 7x 3 +7 f 0 (x) = 7(3)x 2 7( 3)x = 21x x 4 (b) Now that we know erivatives, we change our earlier efinition of marginal cost. Before it was either the change in total cost when we increase prouction by one item, or if the total cost was a linear function then it was the slope. From now on: marginal cost is the erivative of the total cost function. Again, like in part (a), we shoul start by rewriting the function in power-ofx-form : C(q) = 165q 1/

3 CHAPTER 3. RULES FOR DERIVATIVES 50 Now we take the erivative C(q) = 165q 1/ MC = C 0 (q) = 165(1/2)q 1/ = 82.5q 1/2 Now we plug in q = 5. Note: you have to take the erivative first, an then plug in q = 5, not the other way aroun. Marginal Cost at q =5 MC(5) C 0 (5) = 82.5(5 1/2 ) = 82.5 p 5 = 36.9 Definition. The tangent line to a function f(x) at a point x = a is given by y = m(x x 0 )+y 0 where x 0 = a, y 0 = f(a), m = f 0 (a). Comments. Some stuents have a har time seeing how to use the formula for the tangent line, at least at first. As an alternative, here s a recipe: Step 0. You are given a function f(x) an a number x = a. Step 1. Plug x = a into f(x) an calculate this number. Step 2. Fin f 0 (x); this is a formula. Step 3. Plug x = a into f 0 (x) an calculate this number. Step 4. Your formula is y = m(x x 0 )+y 0 where x 0 = a, y 0 is the number from step 1 an m is the number from step 3. Example 4. Fin the equation of the tangent line at x = 5 of f(x) =2x 2 x + 3. Graph the results an explain how this confirms we have the correct formula for the erivative. y = m(x x 0 )+y 0 x 0 = a y 0 = f(5) = 2(5) = 48 f 0 (x) =4x 1 m = 4(5) 1 = 19 y = 19(x 5) + 48 Show below is the graph of f(x) an y = 19(x 5) + 48.

4 CHAPTER 3. RULES FOR DERIVATIVES 51 It looks like the line is exactly tanget at x = 5. This confirms that the slope of the line is correct, that it shoul have a slope of 19. This confirms that the formula we use for the slope, f 0 (x) =4x 1 is correct. This confirms that the power rule, x2 =2x is correct. The secon erivative Definition. Recall that we can start with one function, f(x), an efine a secon function f 0 (x), the erivative function of f(x). Since f 0 (x) is a function, we can repeat this process an efine the secon erivative f 00 (x) = the erivative of f 0 (x) = 2 f 2 Now we escribe what the secon erivative tells us graphically: If f 00 (x) > 0thenf is concave up If f 00 (x) < 0thenf is concave own There are four pictures that relate concave up/own to increasing/ecreasing.

5 CHAPTER 3. RULES FOR DERIVATIVES 52 f 0 (x) > 0, f 00 (x) > 0 f 0 (x) < 0, f 00 (x) > 0 f 0 (x) > 0, f 00 (x) < 0 f 0 (x) < 0, f 00 (x) < 0

6 CHAPTER 3. RULES FOR DERIVATIVES 53 Example 5. Let f(x) =x 4 4x 2. Calculate f 0 (x), f 00 (x), an graph f(x), f 0 (x) an f 00 (x). Compare the graph of f 0 (x) to the graph you prouce in Section 2.2 Example 3. See if you can connect the concavity of f(x) to the values of f 00 (x). f(x) =x 4 4x 2 f 0 (x) = (x4 4x 2 ) =4x 3 8x f 00 (x) = (4x3 8x) = 12x 2 8

7 Here are the graphs: We start by ientifying what the concavity is at 5 crucial parts of the original graph.

8 CHAPTER 3. RULES FOR DERIVATIVES 55 Now we relate those spots where we figure out the concavity, into 5 points on the graph of f 00 (x). Remember: the concavity on the graph of f(x) is picture as y-values on the graph of f 00 (x). f 00 (x)

9 CHAPTER 3. RULES FOR DERIVATIVES 56 Example 6. Shown below is a graph of f(x): Fill in the chart below with 0, + or for each entry. For example, put 0 in the first blank spot if you think that f equals 0 at point A. Be prepare to iscuss your answers. f f 0 f 00 A B C D f f 0 f 00 A 0 + B + 0 C + D + + Justifications: f = at A because the y-value there is negative. f = + at B because the y-value there is positive. f = + at C because the y-value there is positive. f = at D because the y-value there is negative. f 0 = 0 at A because the slope there is flat. f 0 = 0 at B because the slope there is flat. f 0 = at C because the slope there is negative f 0 = + at D because the slope there is positive. f 00 = + at A because the curvature there is positive/up. f 00 = at B because the curvature there is negative/own. f 00 = at C because the curvature there is negative/own. f 00 = + at D because the curvature there is positive/up This is where we ene on Wenesay, February 20.