Section 11.3 Rates of Change:

Transcription

1 Section 11.3 Rates of Change: 1. Consider the following table, which describes a driver making a 168-mile trip from Cleveland to Columbus, Ohio in 3 hours. t Time (in hours) f(t) Distance (in miles) (a) What does f(2) = 118 mean in the context of the problem? (b) What does f(3) f(2) mean in the context of the problem? (c) Calculate the average speed with the time intervals stated below: Time Interval Average Speed [2, 3] [1, 3] [0, 2.5] [0.5, 1] [a, b] 12

2 2. Quantity Meaning for the Distance Function Meaning for an Arbitrary Function f b a Time interval = change in time from Change in x from x = a to x = b x = a to x = b f(b) f(a) Distance traveled = corresponding Corresponding change in f(x) as x change in distance as time changes changes from a to b f(b) f(a) b a from a to b Average speed = average rate of Average rate of change of f(x) with change of distance with respect to respect to x as x changes from a to b time as time changes from a to b (where a < b) 3. Graphing Average Rate of Change: 4. If f(x) = x 2 + 4x + 5, find the average rate of change of f(x) with respect to x as x changes from 2 to 3. 13

3 5. Suppose D(t) is the value of the Dow Jones Industrial Average (DJIA) at time t. Approximate the average rate of change of the DJIA with respect to time over the given intervals. (a) From end of 2000 to end of 2006 (b) From end of 2000 to end of 2002 (c) From end of 2000 to end of Suppose a car is stopped at a traffic light. When the light turns green, the car begins to move along a straight road. Assume that the distance traveled by the car is given by the function s(t) = 2t 2 (0 t 30), where t is measured in seconds and the distance, s(t) is the distance in feet. Find the average speed for each time interval below: Time Interval Average Speed Time Interval Average Speed [10, 20] [10, 10.1] [10, 15] [10, 10.01] [10, 11] [10, ] [10, 10.5] [10, 10 + h] 14

4 7. Velocity: If an object moves along a straight line, with position s(t) at time t, then the velocity of the object at t = a is s(a + h) s(a) lim, h 0 h provided that this limit exists. 8. The distance, in feet, of an object from a starting point is given by s(t) = 2t 2 5t + 40, where t is time in seconds. (a) Find the average velocity of the object from 2 to 4 seconds. (b) Find the instantaneous velocity at 4 seconds. 9. The velocity of blood cells is of interest to physicians; a velocity slower than normal might indicate a constriction, for example. Suppose the position of a red blood cell in a capillary is given by s(t) = 1.2t + 5, where s(t) gives the position of a cell in millimeters from some reference point and t is time in seconds. Find the velocity of this cell at time t = a. 15

5 10. Instantaneous Rate of Change: The instantaneous rate of change for a function f when x = a is f f(a + h) f(a) (a) = lim, h 0 h provided that this limit exists. Note, R (x), C (x) and P (x) are also called, marginal revenue, marginal cost and marginal profit. 11. A company determines that the cost (in hundreds of dollars) of manufacturing x cases of computer mice is C(x) = 0.2x 2 + 8x + 40 (0 x 20). (a) Find the average rate of change of cost for manufacturing between 5 and 10 cases. Use units in your answer! (b) Find the instantaneous rate of change with respect to the number of cases produced when 5 cases are produced. Explain what this means in the context of the problem. 16

6 Section 11.4 Tangent Lines and Derivatives: 1. Graph of the Secant Lines transforming into the Tangent line. Recall, the instantaneous rate of change for f(x) at a: f f(a + h) f(a) (a) = lim, h 0 h This is also the slope of the tangent line of the graph of f(x) at the point (a, f(a)). This is also called the derivative of f(x). Note, without the limit this is the slope of the secant line connecting the points x = a + h and x = a on the graph of f(x). 2. Tangent Line: Note, the tangent line to the graph of a function f(x) at x = a is a good approximation of the function f(x) near x = a. This is useful if the function you are looking at is complicated to evaluate, you can always evaluate a line y = mx + b at a point x. (This will also be useful when we look at multi variables.) 3. Graph of places where the derivative does not exist. 17

7 4. Consider y = x (a) Find the slope of the tangent line to the graph when x = 1. (b) Find the equation of the tangent line. 5. Let a be any real number. Find the equation of the tangent line to the graph of f(x) = 7x + 3 at the point when x = a. (a) Is f (3) positive or negative? 6. (b) Which is larger, f (1) or f (5)? (c) For which values of x is f (x) positive? 18

8 7. A student brings a cold soft drink to a 50-minute math class but is too busy during class to drink it. If C(t) represents the temperature of the soft drink (in Fahrenheit) t minutes after the start of class, interpret the meaning of the following statements, including units. (a) C(0) = 38 (b) C(50) C(0) = 30 (c) C(50) C(0) 50 0 = 0.6 (d) C (0) = Let f(x) = 1 x find f (x) 9. Let f(x) = x find f (x) 19

9 10. Let f(x) = x 2 4x (a) Find f (x) (Note, if you ever see a 4-step process in homework, do not do this, follow how we solve this problem!) (b) Calculate and interpret f (1) (c) Find the equation of the tangent line to the graph of f(x) at the point where x = 1. 20

10 11. A sales representative for a textbook-publishing company frequently makes a 4-hour drive from her home in a large city to a university in another city. Let s(t) represent her position t hours into the trip (where position is measured in miles on the highway, with position 0 corresponding to her home). Then s(t) = 5t 3 +30t 2. How would you answer the following questions. (I do not care about the values, just the work for this example.) (a) How far from home will she be after 1 hour? after 1.5 hours? (b) How far apart are the two cities? (c) What is her velocity 1 hour into the trip? 1.5 hours? 12. Approximating the derivative: If $15,000 is deposited in a money market account that pays 1.2% per year compounded monthly, then the amount in the account after n months is ( A(n) = 15, ) n. 12 Approximate A (120) and interpret what this means. 21

11 Section 11.5 Techniques for Finding Derivatives: 1. Notation: f (x), y, dy dx, d dx f(x), D xy, D x [f(x)] 2. Derivative Rules: Let k be a constant 1. If f(x) = k then f (x) = If f(x) = x n then f (x) = nx n If f(x) = k g(x) then f (x) = k g (x). 4. If f(x) = u(x) ± v(x) then f (x) = u (x) ± v (x). 3. Some examples: Find f (x) where f(x) is defined below. (a) f(x) = 5. (b) f(x) = π. (c) f(x) = x. (d) f(x) = x 3. (e) f(x) = 3x 2. (f) f(x) = x. (g) f(x) = 3 x. (h) f(x) = 1 x. (i) f(x) = x x x 8. (j) f(x) = x2 + 4x + 1 x 2 (k) f(x) = x(x 2 + 5x) 22

12 4. Marginal: If C(x) is the cost function, then the marginal cost (rate of change of cost) is given by the derivative C (x) which is the approximate cost of making one more item after x items have been made. 5. Average Cost: If C(x) is the cost, average cost is C(x) = C(x) x. 6. Revenue: If a demand function p = D(q) is given, where p is price and q is quantity, the total revenue is R(q) = q p = q D(q). 7. Let a demand function be p = 500, 000 q, 25, 000 and cost be C(q) = q for selling q items of something (very exciting). Using this information answer the following questions: (a) Find and simplify the revenue function. (b) Find and simplify the marginal revenue function. (c) Find R (10, 000) and explain what this means. 23

13 (d) Find and simplify the profit function. (e) Find and simplify the marginal profit function. (f) Find and simplify the average cost function. (g) Find the marginal average cost. (h) What is the average cost of producing this very exciting item, as production grows without bound? 8. Word problems: What can the problem say which indicates we want to find the derivative?

14 9. Suppose the total cost in hundreds of dollars to produce x thousand barrels of a beverage is given by C(x) = 5x 3 10x , find the marginal cost when x = 100. And explain what this means. 10. The number of Americans (in thousands) who are expected to be over 100 years old can be approximated by the function f(t) = t t t , where t is the year, with t = 0 corresponding to 2000 and 0 t 50. (a) Find a formula giving the rate of change of the number of Americans over 100 years old. (b) Find the rate of change in the number of Americans who are expected to be over 100 years old in the year (c) What is the number of Americans who are expected to be over 100 years old in the year 2015? 25

15 Section 11.6 Derivatives of Products and Quotients: Derivative rules are getting more complicated. For most of these problems, the only calculus we are doing is in the first step. The rest of the problem we are using algebra to simplify. If you are struggling with simplifying the answers, make sure that you either get the help, or practice you need to do well. (Algebra is a must in this section, if you have poor algebra skills you need to find time to work on them.) 1. Product Rule: If f(x) = u(x) v(x) then Quotient Rule: if f(x) = u(x) v(x) then f (x) = u (x) v(x) + v (x) u(x). f (x) = v(x)u (x) u(x)v (x) [v(x)] 2. Nice saying for quotient rule: low d high minus high d low all over the square of what s below. (Remember, you don t want Heidi.) 2. Let f(x) = (2x + 3) (3x 2 ). Find f (x) 3. Let f(x) = (5x 2 +3x)(x 3 +6x+5 x+100) be a profit function. Find the marginal profit function. 26

16 4. Let f(x) = 3x 2 be the population of some animal, at year x. Find the rate of change of this 5 + 2x animal at year x. 5. Find [ ] (5z 3)(2z + 7) D z 3z

17 6. Let g(z) = (2x2 + 3)(5x + 4), find g (z). 6x 7 28

18 7. Let the cost in dollars of manufacturing x hundred small motors be given by Answer the following questions: (a) Find the marginal cost function C(x) = 3x x + 1. (b) Find the average cost per hundred motors. (c) Find the marginal average cost. (d) Find the level of production that minimizes average cost. minimized when the marginal average cost is zero.) (Note: Average cost is generally 29

19 Section 11.7: The Chain Rule Algebra Portion: 1. Let f(x) = 2x 1 and g(x) = 3x + 5. Find the following: (a) g[f(4)] (b) f[g(4)] (c) g[f(x)] (d) f[g(x)] 2. Write h(x) = 2(4x + 1) 2 + 5(4x + 1) as the composition of two functions h(x) = f[g(x)]. 3. Write h(x) = 1 x 2 as the composition of two functions h(x) = f[g(x)]. 30

20 Calculus Portion: 4. Chain Rule: If y is a function of u, say y = f(u) and if u is a function of x, say u = g(x), then y = f(u) = f[g(x)] and dy dx = dy du du dx. If y = f[g(x)], then 5. Find dy dx if y = 3x 2 5x dy dx = f [g(x)] g (x) 6. Let f(x) = (x 2 + 5x) 8 be the function of some population, somewhere. Find the rate of change of the population. 7. Factor the following problems: (a) 4z xz 4 (b) 4(x + 3) (4x)(x + 3) 4 31

21 8. Let P (x) = 4x(3x+5) 5 be the profit function of some company. Find the marginal profit function. 9. Let C(x) = (3x + 2)7 x 1 be the cost function of some company. Find the marginal cost function. 32

22 10. Suppose a sum of $500 is deposited in an account with an interest rate of r percent per year compounded monthly. At the end of 10 years, the balance in the account is given by ( A = r ) Find the rate of change of A with respect to r if r = 5 or 7. And what does this mean? 11. Marginal-Revenue Product: This is an economic concept that approximates the change in revenue when a manufacturer hires an additional employee. Let R(n) = p x where p is price per unit, or the demand function p = f(x) and x is the number of units produced per day. Note, the number of units produced depends on the number of employees, n. So we can really write x as a function of n. dr dn = p dx dn + x dp dn. As we mentioned p can be a function of the number of employees, so we know that If we re-write the first formula we have: dp dn = dp dx dx dn. dr dn = p dx dn + x by factoring out the greatest common factor we have: ( dp dx dx ) dn 12. Find the marginal-revenue product dr dn function is p = 600 x and x = 5n. (in dollars per employee) when n = 20 if the demand 33