Math 215 Homework Set 5: §§15.5 – 15.8 Winter 2009 Most of the following problems are modified versions of problems from your text book Multivariable Calculus by James Stewart. WARNING: Two pages – start early 15.5a. Because you will probably actually need to use these equations at some point in your academic career, please do Problems 45, 52, and 53 of §15.5 of Stewart’s Multivariable Calculus. 15.5b. The temperature at a point (x, y) on the Michigan football field is T(x, y) measured in degrees Fahrenheit. Alan Mitchell runs so that his position after t seconds is given by x = 26 + √ 25 + 3t 2 and y = 14 + t/5 where x and y are measured in yards. The temperature function satisfies T x (36, 15) = .3 and T y (36, 15) = .1. How fast is the temperature rising along Mitchell’s path after 5 seconds? 15.5c. Suppose that the equation G(s, t, u) = 0 implicitly defines each of the three variables s, t, and u as functions of the other two: s = x(t, u), t = y(s, u), and u = z(s, t). If G is differentiable and G s , G t , and G u are all nonzero, show that 1 = − ∂u ∂s · ∂s ∂t · ∂t ∂u . 15.6a. Use the table for wave heights in Problem 4 of §15.3 of Stewart’s Multivariable Calculus to estimate the value of D u f(30, 20), where u = (i + j)/ √ 2. 15.6b. Find the directional derivative of f(x, y, z) = 3x 2 + y 2 + z 2 at the point (5, 3, 4) in the direction of the origin. 15.6c. Show that a differentiable function f decreases fastest at a point P in the direction opposite the gradient vector at P. In which direction is the function f(x, y, z) = x 2 + 3y 2 + 2z 2 decreasing the fastest at the point (5, 3, 4)? 15.6d. After copying the figure from problem 36 of §15.6 in Stewart’s Multivariable Calculus, please do the problem. 15.6e. After copying the figure from problem 38 of §15.6 in Stewart’s Multivariable Calculus, please do the problem. 15.6f. Find the points on the ellipsoid 2x 2 + y 2 + 3z 2 = 6 where the tangent plane is parallel to the plane 4x + 3y + 6z = 13. 15.7a. For most real world work, our observations do not allow us to develop an exact mathematical model. Thus, we look for models which “minimize” the difference between the predicted and the observed. For example, when we observe that our data appears to give a linear relation between two quantities (say temperature and the rate at which crickets chirp (see Problem 22 in §1.2 of Stewart’s Calculus)) it is usually a good idea to use the method of linear regression (also called the method of least squares) to produce a mathematical model which “minimizes” the difference between the observed and the predicted. Do Problem 55 of §15.7 in Stewart’s Multivariable Calculus. The words “minimize” and “minimizes” appear in quotes above because there may be many ways to measure the difference between the observed and the predicted; give some examples of other ways to measure this difference. 15.7b. Find the shortest distance from the point (2,-3,1) to the plane given by the equation 2x−y +z = 7. 15.7c. Consider the function f(x, y) = 3ye x − y 3 − e 3x . Find and classify the critical point of f. Does the function f obtain an absolute extremum at the critical point? You may wish to use MAPLE to help you answer the latter question. 15.7d. You are to design a rectangular building to house the University’s art collection. Per square yard, the cost for the foundation is four times the cost of the material for the walls which is twice the cost of the material used to construct the roof. If the university has D dollars to spend and the cost of the material for the roof is d dollars per square foot, give the dimensions (in yards) which maximize the volume of the building. 15.7e. Find the extreme values for the function f(x, y) = 2x+y 2 −e x on the unit disc D = {(x, y)|x 2 + y 2 ≤ 1}. You may wish to use MAPLE to solve some of the relevant equations. 15.8a. Find the extreme values for the function 2y 2 + 3x 2 −4y −2 on the set {(x, y)| x 2 + y 2 ≤ 25}. 15.8b. Do Problem 65 on page 983 of Stewart’s Multivariable Calculus. Stephen Debacker Untitled