Math 215Homework Set 1: §§13.1–13.3Fall 2009Most of the following problems are modified versions of the problems from your text book, MultivariableCalculus, 6th ed., by James Stewart. Your solution to each problem should be complete, show all work,and be written in complete sentences where appropriate. For Maple problems, include a print-out thatshows all of the work and graphs that you generated in Maple to solve the problem, in addition to anywork you may have done by hand.13.1.1: Find the equation of the sphere that contains the points (−3, 0, 0), (3, 4, 4) and (−4, 3, 4), if itscenter lies in the xz-plane.13.1.2: Sketch and describe in words the region in R3represented by the equationx2+ y2+ z > 2x.13.1.3: Find the volume of the solid that lies inside both of the spheresx2+ y2+ z2= 4andx2+ y2+ z2+ 4x −2y + 4z + 5 = 0.13.2.1: Consider the vectors a = h−1, 0, 2i and b = h1, −1, 3i. Sketch each of the following quantities:(a) a − 2 b(b) −2a + 3 b13.2.2: Problem #30 from §13.2, but take the wind speed to be 100 km/h and the the plane’s airspeed to bethat of a Boeing 737, about 800 km/h.13.2.3: Problem #36 from §13.2.13.3.1: We define the orthogonal projection of a vector b (onto another vector a) to be orthab = b −projab.(a) Show that orthab is orthogonal to a.(b) Under what circumstances is compab = compba?(c) Under what circumstances is projab = projba?(d) Under what circumstances is orthab = orthba?13.3.2: Use scalar projection to show that the distance from a point (x1, y1) to the line ax + by + c = 0 is|ax1+ by1+ c|√a2+ b2.Math 215Homework Set 2: §§13.4–14.2Fall 2009Most of the following problems are modified versions of the problems from your text book, MultivariableCalculus, 6th ed., by James Stewart. Your solution to each problem should be complete, show all work,and be written in complete sentences where appropriate. For Maple problems, include a print-out thatshows all of the work and graphs that you generated in Maple to solve the problem, in addition to anywork you may have done by hand.13.4.1: Problem #16 from §13.4.13.4.2: Find two unit vectors orthogonal to both i + 2j + k and 2j + k. Are there any others? Explain.13.4.3: Let P be a point not on a plane that passes through the points Q, R and S. Let a =−−→QR, b =−→QS,and c =−−→QP . Show that the distance d from P to the plane isd =|(a × b) · c||a × b|.13.5.1: Find an equation for the set of all points that are equidistant from the points (1, 0, −2) and (3, 4, 0).13.5.2: Find the distance between the skew lines with parametric equations x = 1 + t, y = 1 + 6t, z = 2tand x = 1 + 2s, y = 5 + 15s, z = −2 + 6s.14.1.1: Sketch by hand the curve of intersection of the circular cylinder x2+ z2= 4 and the paraboliccylinder z = y2. Then find parametric equation for this curve and use these equations to graph thecurve using Maple.14.2.1: If r(t) 6= 0, show thatddt|r(t)| =1|r(t)|r(t) · r0(t). [Hint: |r(t)|2= r(t) · r(t)]M.1: Maple problem 1. Consider the surface z = xy and the cylinder x2+ y2= 1.(a) Find the vector function r(t) that describes the intersection of these two surfaces.(b) At what points is the tangent to r(t) horizontal?(c) Find the equations of the lines that are tangent to r(t) at these points.(d) Use Maple to plot the two surfaces, their intersection, and the tangent lines. When you have aplot that clearly shows all of these, print it out and submit it with your homework.Math 215Homework Set 3: §§14.3–15.4Fall 2009Most of the following problems are modified versions of the problems from your text book, MultivariableCalculus, 6th ed., by James Stewart. Your solution to each problem should be complete, show all work,and be written in complete sentences where appropriate. For Maple problems, include a print-out thatshows all of the work and graphs that you generated in Maple to solve the problem, in addition to anywork you may have done by hand.15.1.1: Sketch by hand the graph of the plane x + y + z = 3. Draw the circle of radius one lying onthe plane and centered at (1, 1, 1). What is the radius of the largest circle lying on the plane andcentered at (1, 1, 1) that is wholly contained in the first octant? Explain.15.1.2: Use Maple to investigate the family of surfaces z = x2+y2+cxy. In particular, find what values ofc are transitional values at which the surface changes from one type of quadratic surface to another.Explain what surfaces occur on either side of the transition point.15.3.1: You are told that there is a function f whose partial derivatives are fx(x, y) = x+4y and fy(x, y) =3x − y. Do you believe it? If so, what is the function f? If not, explain.15.3.2: Problem #88 from §15.3.15.4.1: Consider the surface z = e−xy/10(√x +√y). Find the equation of the plane tangent to the surfaceat the point (1, 1, 2 e−0.1). Use Maple to graph the surface and the tangent plane, first on thedomain [0, 2] × [0, 2] and then on a domain such that the tangent plane and the surface becomeindistinguishable. Explain what domains you use, and include graphs that clearly show these twocases.15.4.2: Problem #40 from §15.4.15.4.3: Problem #42 from §15.4.M.2: Maple problem 2. Use Maple to graph the three-dimensional surface and the level curves for eachof the following functions.(a) f(x, y) =px2+ y2,(b) g(x, y) = e−√x2+y2, and(c) h(x, y) = sin(px2+ y2).Explain how the graphs of f, g and h are related to the function p(x) =√x. Pick one of yourgraphs to illustrate this relationship; add to it a graph of a vector function r(t) whose z-componentis appropriately related to p(x), and hand the graph in with your homework. (Your explanation ofthe relationship between f, g and h and p should, of course, refer to the graph of r(t).)Math 215Homework Set 4: §§15.5–15.7Fall 2009Most of the following problems are modified versions of the problems from your text book, MultivariableCalculus, 6th ed., by James Stewart. Your solution to each problem should be complete, show all work,and be written in complete sentences where appropriate. For Maple problems, include a print-out thatshows all of the work and graphs that you generated in Maple to solve the problem, in addition to anywork you may have done by hand.15.5.1: Problem #37 from §15.5. Be sure to carefully explain how you obtain your answer.15.5.2: Problem #54 from §15.5.15.6.1: Let g(x, y) = x2− y2+ 4xy. Find the equation(s) of the level curve with g(x, y) = 4. [Hint: youcan solve for y if you want to