Section 3.1 Relative extrema and intervals of increase and decrease.

Transcription

1 Section 3.1 Relative extrema and intervals of increase and decrease. 4 3 Problem 1: Consider the function: f ( x) x 8x 400 Obtain the graph of this function on your graphing calculator using [-10, 10] for the x-axis. Adjust the y-axis accordingly. Sketch a quick picture below. 1. For what intervals is f ( x) increasing? For what intervals is f ( x) decreasing? 2. Label the local extrema and classify as relative maximum or relative minimum. 4 3 Problem 2 part II: Consider the function: f ( x) x 8x 400 Analytically Find: 1. Critical points (possible relative extrema) and approximate their coordinates. 2. For what intervals is f ( x) increasing? For what intervals is f ( x) decreasing? 3. Label the local extrema and classify as relative maximum or relative minimum. 1

2 Desmos graph for derivative If f(x) is increasing, then f (x) = + If f(x) is decreasing, then f (x) = If f (x) = 0 or f (x) = undefined, then x = a critical number. What does f (x) tell us about the following graph of f(x)? What does the graph of f (x) look like? x-axis 2

3 Graph the derivative of f(x) and check your answer with a calculator. f(x) = x 3 6x 2 + 5x f (x) = Problem 1: f(x) = x 3 4x Domain: 2. Critical points (possible relative extrema) and approximate their coordinates. 3. For what intervals is f ( x) increasing? For what intervals is f ( x) decreasing? 4. Label the local extrema and classify as relative maximum or relative minimum. 3

4 Problem 2: f(x) = 1 4 t 2 1. Domain (verify with the calculator): 2. Critical points (possible relative extrema) and approximate their coordinates. 3. For what intervals is f ( x) increasing? For what intervals is f ( x) decreasing? 4. Label the local extrema and classify as relative maximum or relative minimum. 4

5 Problem 3: f(x) = x x+1 1. Domain (verify with the calculator): 2. Critical points (possible relative extrema) and approximate their coordinates. 3. For what intervals is f ( x) increasing? For what intervals is f ( x) decreasing? 4. Label the local extrema and classify as relative maximum or relative minimum. 5

6 Try these: Problem 3: f(x) = x Domain (verify with the calculator): 2. Critical points (possible relative extrema) and approximate their coordinates. 3. For what intervals is f ( x) increasing? For what intervals is f ( x) decreasing? 4. Label the local extrema and classify as relative maximum or relative minimum. 5. For the following graph, for what values of x is: a) f (x) = + b) f (x) = c) f (x) = 0 (1,10) Application: (5, 2) 6

7 Problem 4: To produce x units of a particular commodity, a monopolist has a total cost of C(x) = 2x 2 + 3x + 5 The price at which x units will be sold is p(x) = 5 2x a) Find the profit function. b) Over what interval are the profits increasing? What is happening to the price over this interval? c) For what level of production is the profit maximized? d) For what price is the profit maximized? e) What is the maximum profit? 7

8 Section 3.2 Concavity and Points of Inflection. Desmos graph for derivative Warm-up: This is the graph of y f () x By studying the graph of y f () x above, determine the sign of f() x, f '( x ), and f ''( x ) for each of the specified x values. x f() x f f '( x ) ''( x ) 8

9 f (x) = + f (x) = f (x) = 0 or doesn t exist f(x) is increasing f(x) is decreasing Critical number, possible extrema f (x) = + f (x) = f (x) = 0 or doesn t exist f (x) is increasing, f(x) is concave up. f (x) is decreasing, f(x) is concave down. Possible point of inflection. Problem 1: Find the intervals of concavity for f(x) = 2x 6 5x 4 + 7x 3 and all points of inflection. 9

10 Problem 2: Find the intervals of concavity for f(x) = 3x 5 5x 4 1 and all points of inflection. Problem 3: Find the intervals of concavity for f(x) = (x 4) 5 3 and all points of inflection. 10

11 Using the second derivative to find relative extrema. For f (critical point) = 0 f (critical point) = + f (critical point) = Problem 4. a) Find the critical points of f(x) = 2x 3 + 3x 2 12x 7. b) Use the second derivative to test and classify the critical points as a relative max or min. Problem 5. a) Find the critical points of f(x) = x + 1 x Domain: b) Use the second derivative to test and classify the critical points as a relative max or min. 11

12 Application-Point of Diminishing Return Rates are going down Rates are going up Point of diminishing return. Maximum rate of change. An efficiency study of the afternoon shift at a factory indicates that an average worker who arrives on the job at noon and leaves at 5pm will have produced Q(t) = t 3 + 6t t, for 0 t 5 units t hours later. a) Find the workers efficiency rate of production. b) Find the value of t where the workers efficiency rate is maximized. This is called the point of diminishing return. c) At what time during the afternoon shift is the worker performing least efficiently? Hint: Least possible rate for the given interval. Hint: The shift starts at noon, hits a max rate at, and then goes down towards 5pm. 12

13 A company estimates that if x thousand dollars are spent on marketing a certain product, then S(x) units of the product will be sold each month, where a) Find the Rate of change for Sales of the product. S(x) = x x x b) Find the maximum sales of the product and the amount of money spend on advertising to obtain the maximum sales. c) Find the amount of money spent on advertising when the sales are increasing most rapidly. 13

14 Section 3.3 Curve sketching. 4 3 Problem I: Consider the function: f ( x) x 8x 400 f (x) = f (x) = Find the relative extrema and intervals of increase and decrease. Find the points of inflection and concavity. 14

15 GRAPHING RATIONAL FUNCTIONS FIND THE HORIZONTAL ASYMPTOTES (the functions limit, what f(x) is approaching) m= The degree of numerator, n= The degree of denominator 1) if m=n, then the horizontal asymptote is: y = 2) if m<n, then the horizontal asymptote is: y=0, the x-axis 3) if m>n, and m-n=1, then the function has a slant asymptote. Leading coefficient of numerator Leading coefficient of denominator Problem II: Consider the function: f(x) = 1 x 2 4 Domain: A: f (x) = f (x) = 15

16 Problem III: Consider the function: f(x) = 3x2 x 2 +2x 15 Domain: A: f (x) = f (x) = 16

17 Problem IV: Consider the function: f(x) = x 1 x Domain: A: f (x) = f (x) = 17

18 Application: A company estimates that if x thousand dollars are spent on the marketing of a certain product, then Q(x) thousand units of the product will be sold, where Q(x) = 7x 27 + x 2 1) Sketch a graph. (calc. to check) 2) For what marketing expenditure x are sales maximized? 3) What is the maximum sales level? 4) For what value of x is the sales rate minimized? Q (x) = Q (x) = 18

19 Section 3.4 Optimization and Elasticity of Demand. Finding absolute Extrema. Example 1: Find the absolute extrema of f(x) = x 2 + 4x + 5 on the interval 3 x 1. Relative extrema and endpoints: f(relative extrema and endpoints): Absolute Max., Min., Neither: Example 2: Find the absolute extrema of f(x) = 10x x x on the interval 0.5 x 2. Relative extrema and endpoints: f(relative extrema and endpoints): Absolute Max., Min., Neither: 19

20 Example 3: Find the absolute extrema of f(x) = t2 Domain: t 1 on the interval 2 x 0. Relative extrema and endpoints: f(relative extrema and endpoints): Absolute Max., Min., Neither: Absolute extrema on an open interval. Example 4: Find the absolute extrema of f(x) = 2x + 32 Domain: x on the open interval x 0. 20

21 Applications. R (q) = C (q) and R (q) < C (q) Example 6: p(q) = 37 2q, total cost. C(q) = 3q 2 + 5q + 75, where p is the price, q the units, and C is the 1. Find the Profit function and the marginal profit. 2. Find the revenue and the marginal revenue. 3. Find the marginal cost 4. Sketch a graph of the profit function, marginal revenue function, and marginal cost function. 5. Find the production level that will maximize profit. Hint: R (q) = C (q) and R (q) < C (q) 6. Find the production level that will maximize profit the old way. 21

22 Example 6 part II: p(q) = 37 2q, the total cost. C(q) = 3q 2 + 5q + 75, where p is the price, q the units, and C is 1. Find the minimum average cost and marginal cost 2. Sketch a graph of the average cost and marginal cost. 3. Find the production level that will minimize average cost. Hint: A(q) = C (q) 4. Take the derivative of average cost and find the production level that will minimize average cost. 5. What is the minimum average cost? 22

23 Try 7: p(q) = 180 2q, cost. C(q) = q 3 + 5q + 162, where p is the price, q the units, and C is the total 7. Find the Profit function and the marginal profit. 1. Find the revenue and the marginal revenue. 2. Find the marginal cost 3. Sketch a graph of the profit function, marginal revenue function, and marginal cost function. 4. Find the production level that will maximize profit. Hint: R (q) = C (q) and R (q) < C (q) 5. Find the production level that will maximize profit the old way. 23

24 Try 7 part II: p(q) = 37 2q, total cost. C(q) = 3q 2 + 5q + 75, where p is the price, q the units, and C is the 6. Find the minimum average cost and marginal cost 7. Sketch a graph of the average cost and marginal cost. 8. Find the production level that will minimize average cost. Hint: AC(q) = C (q) 9. Take the derivative of average cost and find the production level that will minimize average cost. 10. What is the minimum average cost? 24

25 Section 3.4 part II Elasticity of demand. Price elasticity of demand, E(p) is: E(p) = p q dq dp = p q q, where p is the demand price and q is the demanded quantity. % rate of change for q = 100 dq dp q % rate of change for p = 100 dp dp q = 100 p Here is what all of this means: % rate of change for q 100 E(p) = % rate of change for p = 100 p dq dp q = p q dq dp % rate of change of quantity, q E(p) = % rate of change of price, p E E% decrease in quantity q = 1 1% increase in price p Elasticity Inelastic E<1 Unit elasticity E=1 Elastic E>1 1% increase in price Demand decreases less than 1% Demand decreases 1% Demand decreases more than 1% Implies Following A Price Increase Following A Price Decrease R' is + Revenue increases. Revenue decreases. R'=0 Revenue unchanged. Revenue unchanged. R' is - Revenue decreases. Revenue increases When the price of gas in a certain city is $3, then 1,000,000 gallons are sold. What is the revenue? When the price of gas in a certain city is $4, then 775,000 gallons are sold. What is the revenue? By what percent did the price go up, and by what percent did the number of gallons decrease? What is happening to the revenue as the price increases, elastic or inelastic? 25

26 Example 1: Suppose the demand q and price p for a certain commodity are related by the linear equation q = 240 2p for (0 p 120) a) What does the interval, (0 p 120), mean and where did it come from? b) Express the elasticity of demand as a function of p. c) Calculate the elasticity of demand when the price is p=100. Interpret your answer. d) Calculate the elasticity of demand when the price is p=50. Interpret your answer. e) At what price is the demand of unit elasticity? f) At what price is the demand inelastic? g) At what price is the demand elastic? 26

27 Example 1 part II: Suppose the demand q and price p for a certain commodity are related by the linear equation q = 240 2p for (0 p 120) a) At what price is revenue maximized? Hint: find R(p). b) At what price is revenue increasing? c) At what price is revenue decreasing? d) Do you see a relationship between demand elasticity and revenue? e) Sketch a graph of the revenue function, R(p), and label the regions that are inelastic, of unit elasticity, and elastic. 27

28 Review: 1. The formula for calculating E is..... NB: Increasing the price of an item by 1% causes a decrease of..% in the quantity of goods sold. 2. When is the demand elastic? When is the demand inelastic?. 3. Say the price of an item is raised. If demand is inelastic ( E 1), revenue will -crease. If demand is elastic ( E 1), revenue will -crease. (In this case it will be better to the price.) 4. What does it mean if E 2? 5. Say p $20 and q 50 Let E 4. If p increases by 1% to $20.20 then the demand will decrease by.. to Let E 0.5. If p increases by 1% to $20.20 then the demand will increase by.. to 6. Say once again, that p $20 and q 50. The revenue R = If E 4 and p increases to $20.20 the demand decreases to 48. The revenue -creases to.. The price should be. To increase revenue. If E 0.5 and p increases to $20.20 the demand decreases to The revenue.-creases to. The price should be to increase revenue. 28

29 Ex2: When an electronics store prices a certain brand of stereo at p dollars per set, it is found that q sets will be sold each month, where q 2 3pq 22. (a) Find the elasticity of demand for the stereos. Hint: implicit differentiation solve for dq/dp (b) For a unit price of p $3, is the demand elastic, inelastic, or of unit elasticity. Interpret your answer. 29

30 Section 3.5 Additional Applied Optimization Hw :1-33 eoo. 1) A dairy farmer plans to enclose a rectangular pasture adjacent to a river. To provide enough grass for the herd, the pasture must contain 5,000 square yards. No fencing is required along the river. What dimensions will use the least amount of fencing? (minimize perimeter) 30

31 Try: A cable is to be run from a power plant on one side of a river 900 meters wide to a factory on the other side, 3000 meters downstream. The cost of running the cable under the water is $5 per meter, while the cost over land is $ 4 per meter. What is the most economical route over which to run the cable? y x 900 m 3000 m 31

32 More optimization for profit, revenue, and cost. Mateo owns a small company that makes souvenir T-shirts. He can produce the shirts at a cost of $2 apiece. The shirts have been selling for $5 apiece, and at this price, tourists have been buying 4,000 shirts a month. Mateo plans to raise the price of shirts and expects that for each $1 increase in price, 400 fewer shirts will be sold each month. a) Assuming a linear function, form the demand function in terms of q, the number of units sold. b) What price should Mateo charge to maximize revenue? c) What price should Mateo charge per shirt to maximize profit? 32

33 Try: Farmers can get $8 per bushel for their potatoes on July 1, and after that, the price drops 8 cents per bushel per day. On July 1, a farmer has 80 bushels of potatoes in the field and estimates that the crop is increasing at a rate of 1 bushel per day. When should the farmer harvest the potatoes to maximize revenue? Hint: Price function= Quantity function= Revenue= (price function) (quantity function)= 33