EXERCISES Chapter 14: Partial Derivatives. Finding Local Extrema. Finding Absolute Extrema

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1 34 Chapter 4: Partial Derivatives EXERCISES 4.7 Finding Local Etrema Find all the local maima, local minima, and saddle points of the functions in Eercises ƒs, d = ƒs, d = ƒs, d = ƒs, d = ƒs, d = ƒs, d = ƒs, d = ƒs, d = ƒs, d = ƒs, d = ƒs, d = ƒs, d = ƒs, d = ƒs, d = ƒs, d = + ƒs, d = ƒs, d = ƒs, d = ƒs, d = ƒs, d = ƒs, d = >3-4 ƒs, d = ƒs, d = ƒs, d = ƒs, d = ƒs, d = ƒs, d = ƒs, d = ƒs, d = sin 3. ƒs, d = e cos Finding Absolute Etrema In Eercises 3 38, find the absolute maima and minima of the functions on the given domains. 3. ƒs, d = on the closed triangular plate bounded b the lines =, =, = in the first quadrant 3. Ds, d = on the closed triangular plate in the first quadrant bounded b the lines =, = 4, = 33. ƒs, d = + on the closed triangular plate bounded b the lines =, =, + = in the first quadrant 34. s, d = on the rectangular plate 5, s, d = on the rectangular plate 5, ƒs, d = on the rectangular plate, 37. ƒs, d = s4 - d cos on the rectangular plate 3, -p>4 p>4 (see accompaning figure). z (4 ) cos 38. ƒs, d = on the triangular plate bounded b the lines =, =, + = in the first quadrant 39. Find two numbers a and b with a b such that s6 - - d d La has its largest value. 4. Find two numbers a and b with a b such that b b s4 - - d >3 d La has its largest value. 4. emperatures he flat circular plate in Figure 4.46 has the shape of the region +. he plate, including the boundar where + =, is heated so that the temperature at the point (, ) is s, d = + -. Find the temperatures at the hottest and coldest points on the plate. z

2 4.7 Etreme Values and Saddle Points Find the critical point of FIGURE 4.46 Curves of constant temperature are called isotherms. he figure shows isotherms of the temperature function s, d = + - on the disk + in the plane. Eercise 4 asks ou to locate the etreme temperatures. ƒs, d = + - ln in the open first quadrant s 7, 7 d and show that ƒ takes on a minimum there (Figure 4.47). FIGURE 4.47 he function ƒs, d = + - ln (selected level curves shown here) takes on a minimum value somewhere in the open first quadrant 7, 7 (Eercise 4). heor and Eamples 43. Find the maima, minima, and saddle points of ƒ(, ), if an, given that a. ƒ = - 4 and ƒ = - 4 b. ƒ = - and ƒ = - 4 c. ƒ = 9-9 and ƒ = + 4 Describe our reasoning in each case. 44. he discriminant ƒ ƒ - ƒ is zero at the origin for each of the following functions, so the Second Derivative est fails there. Determine whether the function has a maimum, a minimum, or neither at the origin b imagining what the surface z = ƒs, d looks like. Describe our reasoning in each case. a. ƒs, d = b. ƒs, d = - c. ƒs, d = d. ƒs, d = 3 e. ƒs, d = 3 3 f. ƒs, d = Show that (, ) is a critical point of ƒs, d = + k + no matter what value the constant k has. (Hint: Consider two cases: k = and k Z. ) 46. For what values of the constant k does the Second Derivative est guarantee that ƒs, d = + k + will have a saddle point at (, )? A local minimum at (, )? For what values of k is the Second Derivative est inconclusive? Give reasons for our answers. 47. If ƒ sa, bd = ƒ sa, bd =, must ƒ have a local maimum or minimum value at (a, b)? Give reasons for our answer. 48. Can ou conclude anthing about ƒ(a, b) if ƒ and its first and second partial derivatives are continuous throughout a disk centered at (a, b) and ƒ sa, bd and ƒ sa, bd differ in sign? Give reasons for our answer. 49. Among all the points on the graph of z = - - that lie above the plane + + 3z =, find the point farthest from the plane. 5. Find the point on the graph of z = + + nearest the plane + - z =. 5. he function ƒs, d = + fails to have an absolute maimum value in the closed first quadrant Ú and Ú. Does this contradict the discussion on finding absolute etrema given in the tet? Give reasons for our answer. 5. Consider the function ƒs, d = over the square and. a. Show that ƒ has an absolute minimum along the line segment + = in this square. What is the absolute minimum value? b. Find the absolute maimum value of ƒ over the square. Etreme Values on Parametrized Curves o find the etreme values of a function ƒ(, ) on a curve = std, = std, we treat ƒ as a function of the single variable t and

3 36 Chapter 4: Partial Derivatives use the Chain Rule to find where dƒ> dt is zero. As in an other singlevariable case, the etreme values of ƒ are then found among the values at the a. critical points (points where dƒ> dt is zero or fails to eist), and b. endpoints of the parameter domain. Find the absolute maimum and minimum values of the following functions on the given curves. 53. Functions: a. ƒs, d = + b. gs, d = c. hs, d = + i. he semicircle + = 4, Ú ii. he quarter circle + = 4, Ú, Ú Use the parametric equations = cos t, = sin t. 54. Functions: a. ƒs, d = + 3 b. gs, d = c. hs, d = + 3 i. he semi-ellipse s >9d + s >4d =, Ú ii. he quarter ellipse s >9d + s >4d =, Ú, Ú Use the parametric equations = 3 cos t, = sin t. 55. Function: ƒs, d = i. he line = t, = t + ii. he line segment = t, = t +, - t iii. he line segment = t, = t +, t 56. Functions: a. ƒs, d = + b. gs, d = >s + d i. he line = t, = - t ii. he line segment = t, = - t, t Least Squares and Regression Lines When we tr to fit a line = m + b to a set of numerical data points s, d, s, d, Á, s n, n d (Figure 4.48), we usuall choose the line that minimizes the sum of the squares of the vertical distances from the points to the line. In theor, this means finding the values of m and b that minimize the value of the function w = sm + b - d + Á + sm n + b - n d. he values of m and b that do this are found with the First and Second Derivative ests to be () with all sums running from k = to k = n. Man scientific calculators have these formulas built in, enabling ou to find m and b with onl a few ke strokes after ou have entered the data. he line = m + b determined b these values of m and b is called the least squares line, regression line, or trend line for the data under stud. Finding a least squares line lets ou. summarize data with a simple epression,. predict values of for other, eperimentall untried values of, 3. handle data analticall. EXAMPLE Find the least squares line for the points (, ), (, 3), (, ), (3, 4), (4, 5). Solution hen we find We organize the calculations in a table: k k k k k g and use the value of m to find b = n a a k - m a k b, P (, ) P (, ) P n ( n, n ) m b FIGURE 4.48 o fit a line to noncollinear points, we choose the line that minimizes the sum of the squares of the deviations. sds5d - 5s39d m = sd =.9-5s3d k (3) Equation () with n = 5 and data from the table m = a a k ba a k b - n a k k, a a k b - n a k () b = A5 - A.9BABB =.. 5 he least squares line is =.9 +. (Figure 4.49). Equation (3) with n = 5, m =.9

4 4.7 Etreme Values and Saddle Points P (, ) P (, 3) P 4 (3, 4) P 5 (4, 5).9. P 3 (, ) 3 4 FIGURE 4.49 he least squares line for the data in the eample. In Eercises 57 6, use Equations () and (3) to find the least squares line for each set of data points. hen use the linear equation ou obtain to predict the value of that would correspond to = s -, d, s, d, s3, -4d 58. s -, d, s, d, s, 3d 59. (, ), (, ), (, 3) 6. (, ), (, ), (3, ) 6. Write a linear equation for the effect of irrigation on the ield of alfalfa b fitting a least squares line to the data in able 4. (from the Universit of California Eperimental Station, Bulletin No. 45, p. 8). Plot the data and draw the line. ABLE 4. Crater sizes on Mars >D (for Diameter in left value of km, D class interval) Frequenc, F Köchel numbers In 86, the German musicologist Ludwig von Köchel made a chronological list of the musical works of Wolfgang Amadeus Mozart. his list is the source of the Köchel numbers, or K numbers, that now accompan the titles of Mozart s pieces (Sinfonia Concertante in E-flat major, K.364, for eample). able 4.3 gives the Köchel numbers and composition dates () of ten of Mozart s works. a. Plot vs. K to show that is close to being a linear function of K. b. Find a least squares line = mk + b for the data and add the line to our plot in part (a). c. K.364 was composed in 779. What date is predicted b the least squares line? ABLE 4.3 Compositions b Mozart Köchel number, Year composed, K ABLE 4. Growth of alfalfa (total seasonal depth (average alfalfa of water applied, in.) ield, tons/acre) Craters of Mars One theor of crater formation suggests that the frequenc of large craters should fall off as the square of the diameter (Marcus, Science, June, 968, p. 334). Pictures from Mariner IV show the frequencies listed in able 4.. Fit a line of the form F = ms>d d + b to the data. Plot the data and draw the line. 64. Submarine sinkings he data in able 4.4 show the results of a historical stud of German submarines sunk b the U.S. Nav during 6 consecutive months of World War II. he data given for each month are the number of reported sinkings and the number of actual sinkings. he number of submarines sunk was slightl greater than the Nav s reports implied. Find a least squares line for estimating the number of actual sinkings from the number of reported sinkings.

5 38 Chapter 4: Partial Derivatives ABLE 4.4 Sinkings of German submarines b U.S. during 6 consecutive months of WWII Guesses b U.S. (reported sinkings) Actual number Month COMPUER EXPLORAIONS Eploring Local Etrema at Critical Points In Eercises 65 7, ou will eplore functions to identif their local etrema. Use a CAS to perform the following steps: a. Plot the function over the given rectangle. b. Plot some level curves in the rectangle. c. Calculate the function s first partial derivatives and use the CAS equation solver to find the critical points. How do the critical points relate to the level curves plotted in part (b)? Which critical points, if an, appear to give a saddle point? Give reasons for our answer. d. Calculate the function s second partial derivatives and find the discriminant ƒ ƒ - ƒ. e. Using the ma-min tests, classif the critical points found in part (c). Are our findings consistent with our discussion in part (c)? 65. ƒs, d = + 3-3, -5 5, ƒs, d = 3-3 +, -, ƒs, d = , -3 3, ƒs, d = , -3> 3>, -3> 3> ƒs, d = , -4 3, - ƒs, d = e 5 ln s + d, s, d Z s, d, s, d = s, d, -, -