Lecture 26: Section 5.3 Higher Derivatives and Concavity

Transcription

1 L26-1 Lecture 26: Section 5.3 Higher Derivatives and Concavity ex. Let f(x) = ln(e 2x + 1) 1) Find f (x). 2) Find d dx [f (x)].

2 L26-2 We define f (x) = Higher Order Derivatives For y = f(x), we can write The first derivative: f (x) = dy dx = d dx [f(x)] (1) The second derivative: (2) The third derivative: (n) The nth derivative:

3 L26-3 Application: Second Derivative and Rate of Change We have seen that the derivative tells us the rate of change of one quantity with respect to another. It can be important to know how that rate of change is itself changing: that is, what is the rate of change of the first derivative? We can answer that question with the second derivative. ex. The number of unemployed workers N(t) in a certain community can be given by the formula N(t) = t 3 + 9t t , where t is the number of months after the main employer, a furniture manufacturing firm, stopped production in order to move the plant overseas. ex. How many workers were unemployed when production ceased?

4 L26-4 ex. After how many months was the maximum number of workers unemployed? ex. In order to help the community, state government provided funds to hire workers for road construction projects. Due to limited resources, that funding was designed to phase out when the rate of unemployment began to decline. When did that occur?

5 L26-5 Velocity and Acceleration Recall the following ideas: Position Function s(t) NOTE: h(t) Instantaneous velocity: Acceleration

6 L26-6 ex. Suppose that an object is moving along a straight line so that its position in feet from a starting point is given by s(t) = 2t 3 21t t where t is measured in seconds. 1. Find the velocity function v(t). 2. When is the object moving forwards and when is it moving backwards?

7 L Find the function a(t) which gives the acceleration of the object at any time t. When is the acceleration equal to 0? 4. When is the object speeding up and when is it slowing down?

8 L26-8 Concavity We now return to graphing and the shape of a curve. Consider the graph of f(x) = x 3 shown below: Def. Let function f be a function defined on an open interval (a, b). The graph of f is 1) concave upward (concave up) on (a, b) if 2) concave downward on (a, b) if

9 L26-9 ex. Consider the following graph: Find each interval on which f(x) is concave up and down. NOTE: We say that f is concave upward (downward) at a point x = c if

10 L26-10 Test for Concavity Let f be a function whose second derivative exists on interval (a, b). 1) If f (x) > 0 for each value of x in (a, b), then 2) If f (x) < 0 for each value of x in (a, b), then Why? Consider the slope of the tangent lines of f on (a, b):

11 L26-11 To find the intervals on which f is concave upward and downward: 1) Determine the values of x for which f (x) = 0 or where f (x) is not defined. Also consider any discontinuities of f. 2) Set up a number line to find the sign of f on the intervals determined by the x-values from (1), and use the theorem to determine the concavity of f on each of those intervals. ex. Find each interval on which f(x) = 6x 2 x4 4 is concave upward and downward.

12 L26-12 Consider the graph of f(x) = 6x 2 x4 4 : Inflection Points Def. Suppose f is continuous and its graph has a tangent line at x = c. The point (c, f(c)) is a point of inflection if

13 L26-13 Note the following about inflection points: 1) 2) Let (c, f(c)) be a point of inflection of the graph of f. Then either ex. Find each inflection point of f(x) = x 1. On which intervals is f(x) concave up and down? Note that f (x) = 6x 12 x 5. x 3

14 L26-14 ex. Find each inflection point and intervals on which f(x) = x 2/3 (x + 5) is concave up and down, given that f 10x 10 (x) =. 9x 4/3

15 L26-15 Additional Example ex. Suppose that total sales S (in hundreds of dollars) of a product are related to the amount of money x spent on advertising according to the function S(x) = 0.002x x 2 + x where x is measured in thousands of dollars and 0 x 200. Find the inflection point of S. Why is it called the point of diminishing returns?

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17 L26-17 Now you try it! 1. Find all higher order derivatives for f(x) = 2x 4 5x 2 + 3x 1. In general, if polynomial f(x) has degree n, how many unique higher order derivatives does it have? 2. The position of a particle is given by the function s(t) = t 3 9t t where t is measured in minutes and s(t) is measured in yards. When is the particle moving forward? When is it moving backwards? Find the acceleration function a(t). On what intervals is the particle speeding up and slowing down? 3. A particle moves along a path defined by s(t) = (t 3 + 1) where s(t) is the position (in inches from the starting point) of the particle after t seconds. Find a formula for the acceleration at any time t. What is its acceleration after 3 seconds? 4. A drug that stimulates reproduction is introduced into a population of viruses. That population can be modeled by the function P (t) = 30t 2 t , 0 t 20, where P (t) is the population after t minutes. At what value of t is the rate of growth maximized? What feature of the graph occurs at that value? 5. Determine the intervals on which the given functions are concave up/down. List any points of inflection. (a) f(x) = x6 3 5x4 + 2x (b) f(x) = x ln(x) 10x (c) f(x) = 3 4 x1/3 (x 4), given that f (x) = x + 2 3x 5/3 (d) f(x) = xe x2 6. By increasing advertising costs x (in hundreds of dollars) for a product, sales S(x) will grow according to the model S(x) = 0.1x 3 + 6x , for 0 x 40. Show that S(x) is increasing on (0, 40). Then find the point of diminishing returns, where the rate of growth in sales is maximized.