Chapter Nine Chapter Nine

Transcription

1 Chapter Nine Chapter Nine

2 6 CHAPTER NINE ConcepTests for Section 9.. Table 9. shows values of f(, ). Does f appear to be an increasing or decreasing function of? Of? Table (a) Increasing function of ; Increasing function of (b) Increasing function of ; Decreasing function of (c) Decreasing function of ; Increasing function of (d) Decreasing function of ; Decreasing function of (b). The function f is an increasing function of (as is held constant down a column) and a decreasing function of (as is held constant across a row.). The total amount, T, of dollars a buer pas for an automobile is a function of the down pament, P, and the interest rate, r, of the loan on the balance. If the interest rate is held constant, is T an increasing or decreasing function of P? (a) Increasing (b) Decreasing (c) T does not change (d) There is not enough information to tell (b). If the down pament P increases, the total amount that must be paid, T, decreases. You could ask to what the special cases of P = 0 and P = T correspond. 3. The total amount, T, of dollars a buer pas for an automobile is a function of the down pament, P, and the interest rate, r, of the loan on the balance. If the down pament is held constant, is T an increasing or decreasing function of r? (a) Increasing (b) Decreasing (c) T does not change (d) There is not enough information to tell (a). If the interest rate r increases, the total amount that must be paid, T, increases. You might ask to what the special case r = 0 corresponds.

3 CHAPTER NINE 7. The number of songbirds, B, on an island depends on the number of hawks, H, and the number of insects, I. Thinking about the possible relationship between these three populations, decide which of the following depicts it. (I) B increases as H increases and I remains constant. (II) B decreases as H increases and I remains constant. (III) B increases as I increases and H remains constant. (IV) B decreases as I increases and H remains constant. (a) I and III (b) I and IV (c) II and III (d) II and IV (e) None of these (c). Under the assumption that hawks eat songbirds, (II) is true. The answer (c) assumes that the songbirds eat the insects. Some songbirds eat onl seeds, so in that case, if the insects eat the seeds, the correct answer is (d). You might ask for a reason as well.. For a certain function z = f(, ), we know that f(0, 0) = 0 and that z goes up b 3 units for ever unit increase in and z goes down b units for ever unit increase in. What is f(, )? (a) (b) 6 (c) (d) (e) (f) 6 (b). At the point (0, 0), the function value is 0. When the -value goes up b, the function value goes up b 3 = 6. When the -value goes up b, the function value goes down b = 0. We have f(, ) = = Which of the graphs (a) (f) shows a cross-section of f(, ) = 0 + with held fied? (a) (b) (c) (d) (e) (f) (b) and (f). When is fied, we have f(, ) = 0 + C = + (0 + C) = + C where C is an arbitrar constant. The graph of = + C is an upside-down parabola with its verte on the -ais, so the onl cross-sections with fied are graphs (b) and (f).

4 8 CHAPTER NINE ConcepTests for Section 9.. Which of the following terms best describes the origin in the contour diagram in Figure 9.? (a) A mountain pass (b) A dr river bed (c) A mountain top (d) A ridge Figure 9. A dr river bed. Follow-up Question. Imagine that ou are walking on the graph of the function. Describe what ou see and how ou move.. Which of the following is true? If false, find a countereample. (a) The values of contour lines are alwas unit apart. (b) An contour diagram that consists of parallel lines comes from a plane. (c) The contour diagram of an plane consists of parallel lines. (d) Contour lines can never cross. (e) The closer the contours, the steeper the graph of the function. (c) and (e) are true. For (a), Figure 9.3 in the tet, a contour diagram of cardiac outputm is a countereample. For (b), the parabolic clinder, z =, has parallel line contours. For (d), contour lines of the same value can cross, for eample, the contour 0 in z =. Follow-up Question. How would ou have to change the statements to make them true (if possible)? 3. Draw a contour diagram for the surface z =. Choose a point and determine a path from that point on which (a) The altitude of the surface remains constant. (b) The altitude increases most quickl. (a) To maintain a constant altitude, we move along a contour. (b) To find the path on which the altitude increases most quickl, we move perpendicular to the level curves. Find the equations of these paths in the -plane. Choose other paths on the plane and describe them.

5 CHAPTER NINE 9. Figure 9. shows contours of a function. You stand on the graph of the function at the point (,, ) and start walking parallel to the -ais. Describe how ou move and what ou see when ou look right and left Figure 9. You start to the left of the -ais on the contour labeled. At first ou go downhill, and the slope is steep. Then it flattens out. You reach a lowest point right on the ais and start going uphill; the slope is increasing as ou go. To our left and right the terrain is doing the eact same thing; ou are walking across a valle that etends to our right and left. Follow-up Question. What happens if ou walk in a different direction? ConcepTests for Section 9.3 Problems concern the contour diagram for a function f(, ) in Figure 9.3. f(, ) R Q 3 S 7 9 P Figure 9.3. At the point Q, which of the following is true? (a) f > 0, f > 0 (b) f > 0, f < 0 (c) f < 0, f > 0 (d) f < 0, f < 0 (d), since both are negative.. List the points P, Q, R in order of increasing f. (a) P > Q > R (b) P > R > Q (c) R > P > Q (d) R > Q > P (e) Q > R > P (b), since f (P ) > 0, f (R) > 0, and f (P ) > f (R). Also f (Q) < 0.

6 60 CHAPTER NINE 3. Use the level curves of f(, ) in Figure 9. to estimate (a) f (, ) (b) f (, ) (m) Figure 9. (m) (a) We have (b) We have f (, ) f (, ) =.7 = =.. Figure 9. shows level curves of f(, ). At which of the following points is one or both of the partial derivatives, f, f, approimatel zero? Which one(s)? (a) (, 0.) (b) ( 0.,.) (c) (., 0.) (d) ( 0., ) Figure 9. 0

7 (a) At (, 0.), the level curve is approimatel horizontal, so f 0, but f 0. (b) At ( 0.,.), neither are approimatel zero. (c) At (., 0.), neither are approimatel zero. (d) At ( 0., ), the level curve is approimatel vertical, so f 0, but f 0. Ask students to estimate the nonzero partial derivative at one or more of the points. Ask students to speculate about the partial derivatives at (0, 0). CHAPTER NINE 6. Figure 9.6 is a contour diagram for f(, ) with the and aes in the usual directions. At the point P, is f (P ) positive or negative? Increasing or decreasing as we move in the the direction of increasing? 3 P Figure 9.6 At P, the values of f are increasing at an increasing rate as we move in the positive -direction, so f (P ) > 0, and f (P ) is increasing. Ask about how f changes in the -direction and about f. 6. Figure 9.7 is a contour diagram for f(, ) with the and aes in the usual directions. At the point P, if increases, what is true of f (P )? If increases, what is true of f (P )? (a) Have the same sign and both increase. (b) Have the same sign and both decrease. (c) Have opposite signs and both increase. (d) Have opposite signs and both decrease. (e) None of the above. 3 P Figure 9.7 We have f (P ) > 0 and decreasing, and f (P ) < 0 and decreasing, so the answer is (d).

8 6 CHAPTER NINE ConcepTests for Section 9.. Let f(, ) = ln( ). Which of the following are the two partial derivatives of f? Which is which? (a) ln( ) +, (b), ln( ) + (c), + ln( ) 3 (d) + ln( ), (e) + ln( ), (f), ln( ) + 3 The partial derivatives are f (, ) = ln( ) + Thus, the answer is (c), with f first, f second. f (, ) = = ln( ) + =.. Which of the following functions satisf the following equation (called Euler s Equation): f + f = f? (a) 3 (b) + + (c) + (d) Thus Calculate the partial derivatives and check. The answer is (d), and ( ) = = ( ) = = f + f = = = f. 3. Figure 9.8 shows level curves of f(, ). What are the signs of f (P ) and f (P )? f(, ) 3 P Figure 9.8

9 CHAPTER NINE 63 Since f is positive and increasing at P, we have f (P ) > 0. Since f is negative, but getting less negative at P, we have f (P ) > 0. However, f is changing ver slowl, so it would also be reasonable to sa f (P ) 0. As an etension, ask students to find the sign of f (P ).. Figure 9.9 shows level curves of f(, ). What are the signs of f (Q) and f (Q)? f(, ) Q 7 6 Figure 9.9 Since f is positive and decreasing at Q, we have f (Q) < 0. Since f is positive and decreasing at Q, we have f (Q) < 0. However, f is changing ver slowl, so it would also be reasonable to sa f (Q) 0. As an etension, ask students to find the sign of f (Q).. Figure 9.0 shows the temperature T C as a function of distance in meters along a wall and time t in minutes. Choose the correct statement and eplain our choice without computing these partial derivatives. T (a) t (t, 0) < 0 and T t (t, 0) < 0. (b) T t (t, 0) > 0 and T (t, 0) > 0. t T (c) t (t, 0) > 0 and T t (t, 0) < 0. (d) T t (t, 0) < 0 and T (t, 0) > 0. t t (minutes) Figure 9.0 (meters) (c). From the graph of the contour lines, we can see that along the line = 0, the temperature T is increasing, thus T (t, 0) > 0. Since the distance between the contour lines is increasing as time increases along the line = 0, the t rate of change of T t (t, 0) is decreasing, so T (t, 0) < 0. t

10 3 6 CHAPTER NINE ConcepTests for Section 9.. Estimate the global maimum and minimum of the functions whose level curves are in Figure 9.. How man times does each occur? (a) Ma = 6, occurring once; min = 6, occurring once (b) Ma = 6, occurring once; min = 6, occurring twice (c) Ma = 6, occurring twice; min = 6, occurring twice (d) Ma = 6, occurring three times; min = 6, occurring three times (e) None of the above Figure 9. The global ma is about 6, or slightl higher. It occurs three times, twice on the negative -ais and once on the positive -ais. The global min is 6, or slightl lower. It occurs three times, twice on the positive -ais and once on the negative -ais. The answer is (d). Problems refer to the functions f(, ): (a) + 3 (b) + + (c) 3 + (d) cos. Which of these functions has a critical point at the origin? (a). Taking partial derivatives gives f = and f = 6, so = 0 and = 0 give a critical point. The other functions do not have a critical point at the origin. 3. True or False? Function (b) has a local maimum at the origin. False. Since f (0, 0) =, the origin is not a critical point of f, so it cannot be a local maimum.. Which of the functions does not have a critical point? (c). Taking partial derivatives gives f = 3 3 and f = 3 +. Because f is alwas greater than or equal to, there are no critical points.

11 CHAPTER NINE 6 Problems 8 refer to the functions f(, ) : (a) (b) + (c) (/) + (d) 7. Which of the functions has a critical point at the origin? (b). Taking partial derivatives gives f = + = ( + ) and f = + = ( + ), so = 0 and = 0 give a critical point. The others do not have a critical point at the origin. 6. True or false? For the function (b), the second derivative test shows that critical point (0, 0) is a local maimum. False. Taking partial derivatives gives f =, f = 0, and f = +, so so the second derivative test tells us (0, 0) is a saddle point. D = ( 0) 0 ( 0 + ) =, 7. Which of these functions does not have a critical point with = 0? (c). Taking partial derivatives gives f = 3 = ( ) and f = + = ( + ). At critical points, we need = and =, so 3 =, or =. Thus, ma equal either or, so the onl two critical points are (, ) and (, ). 8. Which of these functions does not have a critical point with =? (d). Taking partial derivatives gives f = 3 and f = 7, so the two critical points are = ±7 /, = 0. ConcepTests for Section 9.6. The point P is a maimum or minimum of the function f subject to the constraint g(, ) = + = c, with, 0. For the graphs (a) and (b), does P give a maimum or a minimum of f? What is the sign of λ? If P gives a maimum, where does the minimum of f occur? If P gives a minimum, where does the maimum of f occur? (a) (b) f = 3 f = f = f = f = f = 3 g(, ) = c g(, ) = c P P (a) The point P gives a minimum; the maimum is at one of the end points of the line segment (either the - or the -intercept). The value of λ is negative, since f decreases in the direction in which g increases. (b) The point P gives a maimum; the minimum is at the - or -intercept. The value of λ is positive, since f and g increase in the same direction.

12 66 CHAPTER NINE. Figure 9. shows the optimal point (marked with a dot) in three optimization problems with the same constraint. Arrange the corresponding values of λ in increasing order. (Assume λ is positive.) (a) I<II<III (b) II<III<I (c) III<II<I (d) I<III<II (I) (II) (III) f = 3 f = f = f = 3 f = f = Figure 9. f = 3 f = f = Since λ is the additional f that is obtained b relaing the constraint b unit, λ is larger if the level curves of f are close together near the optimal point. The answer is I<II<III, so (a). For Problems 3, use Figure 9.3. The grid lines are one unit apart. g = c f = f = 3 f = f = f = f = 0 Figure Find the maimum and minimum values of f on g = c. At which points do the occur? The maimum is f = and occurs at (0, ). The minimum is f = and occurs at about (, ).. Find the maimum and minimum values of f on the triangular region below g = c in the first quadrant. The maimum is f = and occurs at (0, ). The minimum is f = 0 and occurs at the origin.

13 CHAPTER NINE 67 Problems 6 concern the following: (a) Maimize subject to + =. (b) Maimize + subject to + =. (c) Minimize + subject to =. (d) Maimize subject to + =.. Which one of (a) (d) does not have its solution at the same point in the first quadrant as the others? The answer is (d). (a) We solve = λ = λ + =, giving =, so =, so (since we are working in the first quadrant) = =. (b) We solve = λ = λ + =, giving =, so =, so (since we are working in the first quadrant) = =. (c) We solve = λ = λ =, giving =, so ( ) =, so (since we are working in the first quadrant) = =. (d) We solve = λ = λ + =. Dividing the first two equations gives =, so (since we are working in the first quadrant) = =. Thus, (a), (b), and (c) have their solution at = =, and (d) has its solution at a different point, namel = =. 6. Which two of (a) (d) have the same etreme value of the objective function in the first quadrant? (a) We have f(, ) = ( ) ( ) =. (b) We have f(, ) = + =. (c) We have f(, ) = ( ) + ( ) =. (d) We have f(, ) = = 6. Thus, (a) and (c) have the same etreme value.

14 68 CHAPTER NINE 7. Find the maimum of the production function f(, ) = in the first quadrant subject to each of the three budget constraints. I. + = II. + = III. 3 + / = Arrange the - and -coordinates of the optimal point in increasing order. Pick one of (a) (e) and one of (f) (j). (a) : I < II < III (f) : I < II < III (b) : III < II < I (g) : III < II < I (c) : II < III < I (h) : II < III < I (d) : II < I < III (i) : II < I < III (e) : III < I < II (j) : III < I < II For (I), we have so = and the solution is = 6, = 6. For (II), we have so =, and the solution is = 3, =.. For (III), we have so 3 = /, and the solution is =, =. Thus, for : III<II<I for : II<I<III The answer is (b) and (i). = λ = λ + =, = λ = λ + =, = 3λ = λ/ 3 + / =,