Unit #18 - Level Curves, Partial Derivatives

Transcription

1 Unit #8 - Level Curves, Partial Derivatives Some problems and solutions selected or adapted from Hughes-Hallett Calculus. Contour Diagrams. Figure shows the densit of the fo population P (in foes per square kilometer) for southern England. Draw two different cross-sections along a north-south line and two different cross-sections along an east-west line of the population densit P. Figure In Problems -6, sketch a contour diagram for the function with at least four labeled contours. Describe in words the contours and how the are spaced.. f(, ) = + 3. f(, ) = Match the pairs of functions (a)-(d) with the contour diagrams (I)-(IV). In each case, show which contours represent f and which represent g. (The - and -scales are equal.) 4. f(, ) = + 5. f(, ) = + (a) f(, ) = +, g(, ) = 6. f(, ) = cos + 7. Match the contour diagrams (a)-(d) with the surfaces (I)-(IV). Give reasons for our choice. (b) f(, ) = + 3, g(, ) = 3 (c) f(, ) =, g(, ) = + ln (d) f(, ) =, g(, ) =

2 Table B Match the surfaces (a)-(e) with the contour diagrams (I)-(V) below. Table C Table D Match each Cobb-Douglas production function (a)-(c) with a graph (I)-(III) and a statement (D)-(G). a. F (L, K) = L 0.5 K 0.5 b. F (L, K) = L 0.5 K 0.5 c. F (L, K) = L 0.75 K 0.75 D. Tripling each input triples output. 0. Match Tables A-D with the contour diagrams (I)-(IV). Table A E. Quadrupling each input doubles output. G. Doubling each input almost triples output.

3 Sketch the contour diagram of each of the following functions.. Below is the contour diagram of f(, ). (a) 3f(, ) (b) f(, ) 0 (c) f(, ) (d) f(, ) Linear Functions 3. The charge, C, in dollars, to use an Internet service is a function of m, the number of months of use, and t, the total number of minutes on-line: C = f(m, t) = m t (a) Is f a linear function? (b) Give units for the coefficients of m and t, and interpret them as charges. (c) Interpret the intercept 35 as a charge. (d) Find f(3, 800) and interpret our answer. Which of the contour diagrams in Problems 4-7 could represent linear functions?

4 (b) Move in the table along a line right one step, up two steps, e.g. from z = 9 ( = 5, = 4) to z = 4 ( = 5, = 6). Then move in the same direction from z = to z = 7. What do ou notice about the changes in z? (c) Show that z = m +n. Use this to eplain what ou observed in parts (a) and (b). Which of the tables of values in Eercises 9 - could represent linear functions? Each column of the table below is linear with the same slope, m = z = 4/5. Each row is linear with the same slope,n = z = 3/. We now investigate the slope obtained b moving through the table along lines that are neither rows nor columns (a) Move down the diagonal of the table from the upper left corner (z = 3) to the lower right corner (z = 3). What do ou notice about the changes in z? Now move diagonall from z = 6 to z = 7. What do ou notice about the changes in z now? Find the linear function whose graph is the plane through the points (4, 0, 0), (0, 3, 0) and (0, 0, ). 4. Find the equation of the linear function z = c+m+n whose graph intersects the z-plane in the line z = and intersects the z-plane in the line z = Find the equation for the linear function described b the table below

5 The Partial Derivative 6. A drug is injected into a patient s blood vessel. The function c = f(, t) represents the concentration of the drug (in mg/l) at a distance mm in the direction of the blood flow measured from the point of injection and at time t seconds since the injection. For the following partial derivatives, What are the units of the following partial derivatives? What are their practical interpretations? What do ou epect their signs to be? (a) c (b) c t 7. The quantit, Q, of beef purchased at a store, in kilograms per week, is a function of the price of beef, b, and the price of chicken, c, both in dollars per kilogram. Time t (months) Conc. c (ppm) The surface z = f(, ) is shown in the graph below. The points A and B are in the -plane. (a) What is the sign of (i) f (A)? (ii) f (A)? (b) The point P in the -plane moves along a straight line from A to B. How does the sign of f (P ) change? How does the sign of f (P ) change? (a) Do ou epect Q b Eplain. (b) Do ou epect Q c Eplain. (c) Interpret the statement Q b quantit of beef purchased. to be positive or negative? to be positive or negative? = 3 in terms of 8. Below is a contour diagram for z = f(, ). Is f positive or negative? Is f positive or negative? Estimate f(, ), f (, ), and f (, ). NOTE: the aes are not positioned in the usual location! Positive values are back and left, and positive values are down and left. This affects our interpretation of the slope. 3. Consider the graph below: 9. An eperiment to measure the toicit of formaldehde ielded the data in the table below. The values show the percent, P = f(t, c), of rats surviving an eposure to formaldehde at a concentration of c (in parts per million, ppm) after t months. Estimate f t (8, 6) and f c (8, 6). Interpret our answers in terms of formaldehde toicit. (a) What is the sign of f (0, 5)? (b) What is the sign of f (0, 5)? 3. The figure below shows a contour diagram for the monthl pament P as a function of the interest rate, r %, and the amount, L, of a 5-ear loan. Estimate P r 5

6 and P at the following points. In each case, give the L units and the everda meaning of our answer. (a) r = 8, L = 4000 (b) r = 8, L = 6000 (c) r = 3, L = 7000 Computing Partial Derivatives Find the partial derivatives in Problems Assume the variables are restricted to a domain on which the function is defined. 33. f and f if f(, ) = (a ) B ( u 0 B ) 36. F v if F = mv r T l 37. if T = π l g 38. f a if f(a, b) = e a sin(a + b) 39. g if g(, ) = ln(e ) 40. Q K if Q = c(a K b + a L b ) γ 4. Mone in a bank account earns interest at a continuous rate, r. The amount of mone, $B, in the account depends on the amount deposited, $P, and the time, t, it has been in the bank according to the formula B = P e rt Find B t terms. and B P and interpret each in financial 4. The Dubois formula relates a person s surface area, s, in m, to weight, w, in kg, and height, h, in cm, b s = f(w, h) = 0.0w 0.5 h 0.75 Find f(65, 60), f w (65, 60), and f h (65, 60). Interpret our answers in terms of surface area, height, and weight. 43. Is there a function f which has the following partial derivatives? If so what is it? Are there an others? f (, ) = f (, ) = 4 3 6