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All Textbook Solutions for Calculus: Early Transcendentals (3rd Edition)
Tangents and normals for an ellipse Consider the ellipse r(t) = 3 cos t, 4 sin t, for 0 t 2. a. Find the tangent vector r, the unit tangent vector T, and the principal unit normal vector N at all points on the curve. b. At what points does |r| have maximum and minimum values? c. At what points does the curvature have maximum and minimum values? Interpret this result in light of part (b). d. Find the points (if any) at which r and N are parallel.51RE52REProperties of space curves Do the following calculations. a. Find the tangent vector and the unit tangent vector. b. Find the curvature. c. Find the principal unit normal vector. d. Verify that |N| = 1 and T N = 0. e. Graph the curve and sketch T and N at two points. 61. r(t)=costi+2costj+5sintk,for0t254REAnalyzing motion Consider the position vector of the following moving objects. a. Find the normal and tangential components of the acceleration. b. Graph the trajectory and sketch the normal and tangential components of the acceleration at two points on the trajectory. Show that their sum gives the total acceleration. 63. r(t) = 2 cos t i + 2 sin t j, for 0 t 2Analyzing motion Consider the position vector of the following moving objects. a. Find the normal and tangential components of the acceleration. b. Graph the trajectory and sketch the normal and tangential components of the acceleration at two points on the trajectory. Show that their sum gives the total acceleration. 64. r(t) = 3t i + (4 t) j + t k, for t 0Analyzing motion Consider the position vector of the following moving objects. a. Find the normal and tangential components of the acceleration. b. Graph the trajectory and sketch the normal and tangential components of the acceleration at two points on the trajectory. Show that their sum gives the total acceleration. 65. r(t) = (t2 + 1) i + 2t j, for t 0Analyzing motion Consider the position vector of the following moving objects. a. Find the normal and tangential components of the acceleration. b. Graph the trajectory and sketch the normal and tangential components of the acceleration at two points on the trajectory. Show that their sum gives the total acceleration. 66. r(t) = 2 cos t i + 2 sin t j + 10t k, for 0 t 259RECurve analysis Carry out the following steps for the given curves C. a. Find T(t) at all points of C. b. Find N(t) and the curvature at all points of C. c. Sketch the curve and show T(t) and N(t) at the points of C corresponding to t = 0 and t = /2. d. Are the results of parts (a) and (b) consistent with the graph? e. Find B(t) at all points of C. f. On the graph of part (c), plot B(t) at the points of C corresponding to t = 0 and t = /2. g. Describe three calculations that serve to check the accuracy of your results in part (a)(f). h. Compute the torsion at all points of C. Interpret this result. 70. C: r(t) = 3 sin t, 4 sin t, 5 cos t, for 0 t 261RE62RE63RE64REFind the domains of f(x, y) = sin xy and g(x, y) = (x2+1)y.Does the graph of a hyperboloid of one sheet represent a function? Does the graph of a cone with its axis parallel to the x-axis represent a function?
Find a function whose graph is the lower half of the hyperboloid −x2 − y2 + z2 = 1.
Can two level curves of a function intersect? Explain.5QC6QC7QC8QCWhat is the domain of the function w = f(x, y, z) = 1/xyz?What is domain of f(x, y) = x2y xy2?What is the domain of g(x, y) = 1/(xy)?What is the domain of h(x,y)=xy?How many axes (or how many dimensions) are needed to graph the function z = f(x, y)? Explain.Explain how to graph the level curves of a surface z = f(x, y).Given the function f(x, y) = 10x+y, evaluate f(2, 1) and f(9, 3).8EThe function z = f(x, y) gives the elevation z (in hundreds of feet) of a hillside above the point (x, y).Use the level curves of f to answer the following questions (see figure). 9. Katie and Zeke are standing on the surface above the point A(2, 2). a. At what elevation are Katie and Zeke standing? b. Katie hikes south to the point on the surface above B(2, 1) and Zeke hikes east to the point on the surface above C(3, 2). Who experienced the greater elevation change and what is the difference in their elevations?The function z = f(x, y) gives the elevation z (in hundreds of feet) of a hillside above the point (x, y). Use the level curves of f to answer the following questions (see figure). 10. Katie and Zeke are standing on the surface above D(1, 0). Katie hikes on the surface above the level curve containing D(1, 0) to B(2, 1) and Zeke walks east along the surface to E(2, 0). What can be said about the elevations of Katie and Zeke during their hikes?Describe in words the level curves of the paraboloid z = x2 + y2.How many axes (or how many dimensions) are needed to graph the level surfaces of w = f(x, y, z)? Explain.The domain of Q = f(u, v, w, x, y, z) lies in n for what value of n? Explain.Give two methods for graphically representing a function with three independent variables.Domains Find the domain of the following functions. 11.f(x, y) = 2xy 3x + 4y16EDomains Find the domain of the following functions. 13.f(x,y)=25x2y2Domains Find the domain of the following functions. 14.f(x,y)=1x2+y225Domains Find the domain of the following functions. 15.f(x,y)=sinxyDomains Find the domain of the following functions. 16.f(x,y)=12y2x2Domains Find the domain of the following functions. 17.g(x,y)=ln(x2y)Domains Find the domain of the following functions. 18.f(x,y)=sin1(yx2)Domains Find the domain of the following functions. 19.g(x,y)=xyx2+y2Domains Find the domain of the following functions. 20.h(x,y)=x2y+4Graphs of familiar functions Use what you learned about surfaces in Sections 13.5 and 13.6 to sketch a graph of the following functions. In each case, identify the surface and state the domain and range of the function 25. f(x, y) = 6 x 2yGraphs of familiar functions Use what you learned about surfaces in Sections 13.5 and 13.6 to sketch a graph of the following functions. In each case, identify the surface and state the domain and range of the function 26. g(x, y) 4Graphs of familiar functions Use what you learned about surfaces in Section 12.1 to sketch a graph of the following functions. In each case identify the surface and state the domain and range of the function. 23.p(x, y) = x2 y2Graphs of familiar functions Use what you learned about surfaces in Section 12.1 to sketch a graph of the following functions. In each case identify the surface and state the domain and range of the function. 22.h(x, y) = 2x2 + 3y2Graphs of familiar functions Use what you learned about surfaces in Section 12.1 to sketch a graph of the following functions. In each case identify the surface and state the domain and range of the function. 25.G(x,y)=1+x2+y2Graphs of familiar functions Use what you learned about surfaces in Section 12.1 to sketch a graph of the following functions. In each case identify the surface and state the domain and range of the function. 24.F(x,y)=1x2y2Graphs of familiar functions Use what you learned about surfaces in Section 12.1 to sketch a graph of the following functions. In each case identify the surface and state the domain and range of the function. 27.P(x,y)=x2+y21Graphs of familiar functions Use what you learned about surfaces in Section 12.1 to sketch a graph of the following functions. In each case identify the surface and state the domain and range of the function. 26.H(x,y)=x2+y2Graphs of familiar functions Use what you learned about surfaces in Sections 13.5 and 13.6 to sketch a graph of the following functions. In each case, identify the surface and state the domain and range of the function 33. g(x, y) 164x2Matching level curves with surfaces Match surfaces af in the figure with level curves AF. (a) (b) (c) (d) (A) (B) (C) (D) (E) (F)Matching surfaces Match functions ad with surfaces AD in the figure. a.f(x, y) = cos xy b.g(x, y) = ln (x2 + y2) c.h(x, y) = 1/(x y) d.p(x, y) = 1/(1 + x2 + y2) (A) (B) (C) (D)Level curves Graph several level curves of the following functions using the given window. Label at least two level curves with their z-values. 30.z = x2 + y2; [4, 4] [4, 4]Level curves Graph several level curves of the following functions using the given window. Label at least two level curves with their z-values. 31.z = x y2; [0, 4] [2, 2]Level curves Graph several level curves of the following functions using the given window. Label at least two level curves with their z-values. 32.z = 2x y; [2, 2] [2, 2]Level curves Graph several level curves of the following functions using the given window. Label at least two level curves with their z-values. 33.z=x2+4y2;[8,8][8,8]Level curves Graph several level curves of the following functions using the given window. Label at least two level curves with their z-values. 34.z=ex22y2;[2,2][2,2]Level curves Graph several level curves of the following functions using the given window. Label at least two level curves with their z-values. 35.z=25x2y2;[6,6][6,6]Level curves Graph several level curves of the following functions using the given window. Label at least two level curves with their z-values. 36.z=yx21;[5,5][5,5]Level curves Graph several level curves of the following functions using the given window. Label at least two level curves with their z-values. 37.z = 3 cos (2x + y); [2, 2] [2, 2]Earned run average A baseball pitchers earned run average (ERA) is A(e, i) = 9e/i, where e is the number of earned runs given up by the pitcher and i is the number of innings pitched. Good pitchers have low ERAs. Assume that e 0 and i 0 are real numbers. a.The single-season major league record for the lowest ERA was set by Dutch Leonard of the Detroit Tigers in 1914. During that season, Dutch pitched a total of 224 innings and gave up just 24 earned runs. What was his ERA? b.Determine the ERA of a relief pitcher who gives up 4 earned runs in one-third of an inning. c.Graph the level curve A(e, i) = 3 and describe the relationship between e and i in this case.Electric potential function The electric potential function for two positive charges, one at (0, 1) with twice the strength as the charge at (0, 1), is given by (x,y)=2x2+(y1)2+1x2+(y+1)2. a.Graph the electric potential using the window [5, 5] [5, 5] [0, 10]. b.For what values of x and y is the potential defined? c.Is the electric potential greater at (3, 2) or (2, 3)? d.Describe how the electric potential varies along the line y = x.Cobb-Douglas production function The output Q of an economic system subject to two inputs, such as labor L and capital K, is often modeled by the Cobb-Douglas production function Q(L, K) = cLaKb, where a. b, and c are positive real numbers. When a + b = 1, the case is called constant returns to scale. Suppose a = 13, b = 23, and c = 40. a.Graph the output function using the window [0, 20] [0, 20] [0, 500]. b.If L is held constant at L = 10, write the function that gives the dependence of Q on K. c.If K is held constant at K = 15, write the function that gives the dependence of Q on L.Resistors in parallel Two resistors wired in parallel in an electrical circuit give an effective resistance of R(x,y)=xx+y, where x and y are the positive resistances of the individual resistors (typically measured in ohms). a.Graph the resistance function using the window [0, 10] [0, 10] [0, 5]. b.Estimate the maximum value of R, for 0 x 10 and 0 y 10. c.Explain what it means to say that the resistance function is symmetric in x and y.Level curves of a savings account Suppose you make a one-time deposit of P dollars into a savings account that earns interest at an annual rate of p% compounded continuously. The balance in the account after t years is B(P, r, t) = Pert, where r = p/100 (for example, if the annual interest rate is 4%, then r = 0.04). Let the interest rate be fixed at r = 0.04 a.With a target balance of 2000, find the set of all points (P, t) that satisfy B = 2000. This curve gives all deposits P and times t that result in a balance of 2000. b.Repeat part (a) with B = 500, 1000, 1500, and 2500, and draw the resulting level curves of the balance function. c.In general, on one level curve, if t increases, does P increase or decrease?Level curves of a savings plan Suppose you make monthly deposits of P dollars into an account that earns interest at a monthly rate of p%. The balance in the account after t years isB(P,r,t)=P((1+r)12t1r), where r = p/100 (for example, if the annual interest rate is 9%, then p = 912 = 0.75 and r = 0.0075). Let the time of investment be fixed at t = 20 years. a.With a target balance of 20,000, find the set of all points (P, r) that satisfy B = 20,000. This curve gives all deposits P and monthly interest rates r that result in a balance of 20,000 after 20 years. b.Repeat part (a) with B = 5000, 10,000, 15,000, and 25,000, and draw the resulting level curves of the balance function.Domains of functions of three or more variables Find the domain of the following functions. If possible, give a description of the domains (for example, all points outside a sphere of radius 1 centered at the origin). 46.f(x, y, z) = 2xy 3xz + 4yzDomains of functions of three or more variables Find the domain of the following functions. If possible, give a description of the domains (for example, all points outside a sphere of radius 1 centered at the origin). 47.g(x,y,z)=1xzDomains of functions of three or more variables Find the domain of the following functions. If possible, give a description of the domains (for example, all points outside a sphere of radius 1 centered at the origin). 48.p(x,y,z)=x2+y2+z29Domains of functions of three or more variables Find the domain of the following functions. If possible, give a description of the domains (for example, all points outside a sphere of radius 1 centered at the origin). 49.f(x,y,z)=yzDomains of functions of three or more variables Find the domain of the following functions. If possible, give a description of the domains (for example, all points outside a sphere of radius 1 centered at the origin). 50.Q(x,y,z)=101+x2+y2+4z2Domains of functions of three or more variables Find the domain of the following functions. If possible, give a description of the domains (for example, all points outside a sphere of radius 1 centered at the origin). 51.F(x,y,z)=yx256EExplain why or why not Determine whether the following statements are true and give an explanation or counterexample. a.The domain of the function f(x, y) = 1 |x y| is {(x, y): x y}. b.The domain of the function Q = g(w, x, y, z) is a region in 3. c.All level curves of the plane z = 2x 3y are lines.Quarterback passer ratings One measurement of the quality of a quarterback in the National Football League is known as the quarterback passer rating. The rating formula is , where c% of a quarterback’s passes were completed, t% of his passes were thrown for touchdowns, i% of his passes were intercepted, and an average of y yards were gained per attempted pass.
In the 2016/17 NFL playoffs, Atlanta Falcons quarterback Matt Ryan completed 71.43% of his passes, 9.18% of his passes were thrown for touchdowns, none of his passes were intercepted, and he gained an average of 10.35 yards per passing attempt. What was his passer rating in the 2016 playoffs?
In the 2016 regular season, New England Patriots quarterback Tom Brady completed 67.36% of his passes, 6.48% of his passes were thrown for touchdowns, 0.46% of his passes were intercepted, and he gained an average of 8.23 yards per passing attempt. What was his passer rating in the 2016 regular season?
If c, t, and y remain fixed, what happens to the quarterback passer rating as i increases? Explain your answer with and without mathematics.
Ideal Gas Law Many gases can be modeled by the Ideal Gas Law, PV = nRT, which relates the temperature (T, measured in Kelvin (K)), pressure (P, measured in Pascals (Pa)), and volume (V, measured in m3) of a gas. Assume that the quantity of gas in question is n = 1 mole (mol). The gas constant has a value of R = 8.3 m3-Pa/mol-K. a.Consider T to be the dependent variable and plot several level curves (called isotherms) of the temperature surface in the region 0 P 100,000 and 0 V 0.5. b.Consider P to be the dependent variable and plot several level curves (called isobars) of the pressure surface in the region 0 T 900 and 0 V 0.5. c.Consider V to be the dependent variable and plot several level curves of the volume surface in the region 0 T 900 and 0 P 100,000.Water waves A snapshot of a water wave moving toward shore is described by the function z = 10 sin (2x 3y), where z is the height of the water surface above (or below) the xy-plane, which is the level of undisturbed water. a.Graph the height function using the window [5, 5] [-5, 5] [15, 15]. b.For what values of x and y is z defined? c.What are the maximum and minimum values of the water height? d.Give a vector in the xy-plane that is orthogonal to the level curves of the crests and troughs of the wave (which is parallel to the direction of wave propagation).Approximate mountains Suppose the elevation of Earths surface over a 16-mi by 16-mi region is approximated by the function z=10e(x2+y2)+5e((x+5)2+(y3)2)/10+4e2((x4)2+(y+1)2). a.Graph the height function using the window [8, 8] [8, 8] [0, 15]. b.Approximate the points (x, y) where the peaks in the landscape appear. c.What are the approximate elevations of the peaks?Graphing functions a.Determine the domain and range of the following functions. b.Graph each function using a graphing utility. Be sure to experiment with the graphing window and orientation to give the best perspective of the surface. 54.g(x,y)=exy63E64EGraphing functions a.Determine the domain and range of the following functions. b.Graph each function using a graphing utility. Be sure to experiment with the graphing window and orientation to give the best perspective of the surface. 57.h(x, y) = (x + y)/(x y)Graphing functions a.Determine the domain and range of the following functions. b.Graph each function using a graphing utility. Be sure to experiment with the graphing window and orientation to give the best perspective of the surface. 58.G(x, y) = ln (2 + sin (x + y))67E68E69E70EPeaks and valleys The following functions have exactly one isolated peak or one isolated depression (one local maximum or minimum). Use a graphing utility to approximate the coordinates of the peak or depression. 63.h(x,y)=1e(x2+y22x)72E73ELevel surfaces Find an equation for the family of level surfaces corresponding to f. Describe the level surfaces. 66.f(x,y,z)=1x2+y2+z2Level surfaces Find an equation for the family of level surfaces corresponding to f. Describe the level surfaces. 67.f(x, y, z) = x2 + y2 zLevel surfaces Find an equation for the family of level surfaces corresponding to f. Describe the level surfaces. 68.f(x, y, z) = x2 y2 zLevel surfaces Find an equation for the family of level surfaces corresponding to f. Describe the level surfaces. 69.f(x,y,z)=x2+2z278EChallenge domains Find the domains of the following functions. Specify the domain mathematically and then describe it in words or with a sketch. 75.f(x,y)=sin1(xy)280E81E82EWhich of the following limits exist?
Give an example of a set that contains none of its boundary points.
Can the limit be evaluated by direct substitution?
What is the analog of the Two-Path Test for functions of a single variable?
5QC1EExplain why f(x, y) must approach a unique number L as (x, y) approaches (a, b) along all paths in the domain in order for lim(x,y)(a,b)f(x,y)to exist.What does it mean to say that limits of polynomials may be evaluated by direct substitution?Suppose (a, b) is on the boundary of the domain of f. Explain how you would determine whether lim(x,y)(a,b)f(x,y) exists.Explain how examining limits along multiple paths may prove the nonexistence of a limit.Explain why evaluating a limit along a finite number of paths does not prove the existence of a limit of a function of several variables.What three conditions must be met for a function f to be continuous at the point (a, b)?Let R be the unit disk {(x, y): x2 + y2 1} with (0, 0) removed. Is (0, 0) a boundary point of R? Is R open or closed?At what points of 2 is a rational function of two variables continuous?10EEvaluate lim(x,y)(5,5)x2y2x+yLet f(x)=x22xy2+1x22xy2+1 Use the Two-Path Test to show that lim(x,y)(1,0)f(x) does not exist. (Hint: Examine lim(x,y)(1,0)(alongy=0)f(x)andlim(x,y)(1,0)(alongx=1)f(x) first)Limits of functions Evaluate the following limits. 11.lim(x,y)(2,9)101Limits of functions Evaluate the following limits. 12.lim(x,y)(1,3)(3x+4y2)Limits of functions Evaluate the following limits. 13.lim(x,y)(3,3)(4x2y2)Limits of functions Evaluate the following limits. 14.lim(x,y)(2,1)(xy83x2y3)Limits of functions Evaluate the following limits. 15.lim(x,y)(0,)cosxy+sinxy2yLimits of functions Evaluate the following limits. 16.lim(x,y)(e2,4)lnxyLimits of functions Evaluate the following limits. 17.lim(x,y)(2,0)x23xy2x+yLimits of functions Evaluate the following limits. 18.lim(u,v)(1,1)10uv2v2u2+v221ELimits at boundary points Evaluate the following limits. 20.lim(x,y)(1,2)y2+2xyy+2xLimits at boundary points Evaluate the following limits. 21.lim(x,y)(3,1)x27xy+12y2x3yLimits at boundary points Evaluate the following limits. 22.lim(x,y)(1,1)2x2xy3y2x+yLimits at boundary points Evaluate the following limits. 23.lim(x,y)(2,2)y24xy2x26ELimits at boundary points Evaluate the following limits. 25.lim(x,y)(1,2)yx+1yx128E29E30ENonexistence of limits Use the Two-Path Test to prove that the following limits do not exist. 29.lim(x,y)(0,0)y42x2y4+x232ENonexistence of limits Use the Two-Path Test to prove that the following limits do not exist. 31.lim(x,y)(0,0)y3+x3xyNonexistence of limits Use the Two-Path Test to prove that the following limits do not exist. 32.lim(x,y)(0,0)yx2y2Continuity At what points of 2 are the following functions continuous? 33.f(x,y)=x2+2xyy3Continuity At what points of 2 are the following functions continuous? 34.f(x,y)=xyx2y2+1Continuity At what points of 2 are the following functions continuous? 35.p(x,y)=4x2y2x4+y2Continuity At what points of 2 are the following functions continuous? 36.S(x,y)=2xyx2y2Continuity At what points of 2 are the following functions continuous? 37.f(x,y)=2x(y2+1)Continuity At what points of 2 are the following functions continuous? 38.f(x,y)=x2+y2x(y21)Continuity At what points of 2 are the following functions continuous? 39.f(x,y)={xyx2+y2if(x,y)(0,0)0if(x,y)=(0,0)Continuity At what points of 2 are the following functions continuous? 40.f(x,y)={y42x2y4+x2if(x,y)(0,0)0if(x,y)=(0,0)Continuity of composite functions At what points of 2 are the following functions continuous? 41.f(x,y)=x2+y2Continuity of composite functions At what points of 2 are the following functions continuous? 42.f(x,y)=ex2+y2Continuity of composite functions At what points of 2 are the following functions continuous? 43.f(x, y) = sin xyContinuity of composite functions At what points of 2 are the following functions continuous? 44.g (x, y) = ln (x y)Continuity of composite functions At what points of 2 are the following functions continuous? 45.h(x, y) = cos (x + y)Continuity of composite functions At what points of 2 are the following functions continuous? 46.p(x, y) = exyContinuity of composite functions At what points of 2 are the following functions continuous? 47.f(x, y) = ln (x2 + y2)Continuity of composite functions At what points of 2 are the following functions continuous? 48.f(x,y)=4x2y2Continuity of composite functions At what points of 2 are the following functions continuous? 49.g(x,y)=x2+y293Continuity of composite functions At what points of 2 are the following functions continuous? 50.h(x,y)=xy4Continuity of composite functions At what points of 2 are the following functions continuous? 51.f(x,y)={sin(x2+y2)x2+y2if(x,y)(0,0)1if(x,y)=(0,0)Continuity of composite functions At what points of 2 are the following functions continuous? 52.f(x,y)={1cos(x2+y2)x2+y2if(x,y)(0,0)0if(x,y)=(0,0)Limits of functions of three variables Evaluate the following limits. 53.lim(x,y,z)(1,ln2,3)zexyLimits of functions of three variables Evaluate the following limits. 54.lim(x,y,z)(0,1,0)(1+y)lnexzLimits of functions of three variables Evaluate the following limits. 55.lim(x,y,z)(1,1,1)yzxyxzx2yz+xy+xzy2Limits of functions of three variables Evaluate the following limits. 56.lim(x,y,z)(1,1,1)xxzxy+yzxxz+xyyzLimits of functions of three variables Evaluate the following limits. 57.lim(x,y,z)(1,1,1)x2+xyxzyzxzLimits of functions of three variables Evaluate the following limits. 58.lim(x,y,z)(1,1,1)xz+5x+yz+5yx+y61EMiscellaneous limits Use the method of your choice to evaluate the following limits. 60.lim(x,y)(0,0)y2x8+y2Miscellaneous limits Use the method of your choice to evaluate the following limits. 61.lim(x,y)(0,1)ysinxx(y+1)Miscellaneous limits Use the method of your choice to evaluate the following limits. 62.lim(x,y)(1,1)x2+xy2y22x2xyy265EMiscellaneous limits Use the method of your choice to evaluate the following limits. 64.lim(x,y)(0,0)|xy|xyMiscellaneous limits Use the method of your choice to evaluate the following limits. 65.lim(x,y)(0,0)|xy||x+y|68EMiscellaneous limits Use the method of your choice to evaluate the following limits. 67.lim(x,y)(2,0)1cosyxy270ELimits of composite functions Evaluate the following limits. 77.lim(x,y)(1,0)sinxyxyLimits of composite functions Evaluate the following limits. 80.lim(x,y)(0,/2)1cosxy4x2y3Limits of composite functions Evaluate the following limits. 79.lim(x,y)(0,2)(2xy)xy74E75E76EPiecewise function Let f(x,y)={sin(x2+y21)x2+y21ifx2+y21bifx2+y2=1. Find the value of b for which f is continuous at all points in 2.78E79E80E81E82ENonexistence of limits Show that lim(x,y)(0,0)axmynbxm+n+cym+n does not exist when a, b, and c are nonzero real numbers and m and n are positive integers.84E85ELimit proof Use the formal definition of a limit to prove that lim(x,y)(a,b)y=b. (Hint: Take = .)Limit proof Use the formal definition of a limit to prove thatlim(x,y)(a,b)(x+y)=a+b. (Hint: Take =/2.)Proof of Limit Law 1 Use the formal definition of a limit to prove thatlim(x,y)(a,b)(f(x,y)+g(x,y))=lim(x,y)(a,b)f(x,y)+lim(x,y)(a,b)g(x,y).Proof of Limit Law 3 Use the formal definition of a limit to prove thatlim(x,y)(a,b)cf(x,y)=clim(x,y)(a,b)f(x,y).Compute fx and fy for f(x, y) = 2xy.Which of the following expressions are equivalent to each other: (a) fxy, (b) fyx, or (c)2fyx? Write2fpq in subscript notation.Compute fxxx and f xxy for f(x, y) = x3y.Compute fxz and fzz for f(x, y, z) = xyz x2z + yz2.Explain why, in Figure 15.33, the slopes of the level curves increase as the pressure increases.
Suppose you are standing on the surface z = f(x, y) at the point (a, b, f(a, b)). Interpret the meaning of fx(a, b) and fy(a, b) in terms of slopes or rates of change.2E3EFind fx and fy when f(x, y) = y8 + 2x6 + 2xy.Find fx and fy when f(x, y) = 3x2y + 2.6EVerify that fxy = fyx. for f(x, y) = 2x3 + 3y2 + 1.Verify that fxy = fyx, for f(x, y) = xey.Find fx,, fy, and fz, for f(x, y, z) = xy + xz + yz.The volume of a right circular cylinder with radius r and height h is V = r2h. Is the volume an increasing or decreasing function of the radius at a fixed height (assume r 0 and h 0)?11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28E29E30EPartial derivatives Find the first partial derivatives of the following functions.
31.
32E33E34E35EMiscellaneous partial derivatives Compute the first part derivatives of the following functions. 66.f(x, y) = 1 cos (2(x + y)) + cos2 (x +y)37E38E39E40E41E42E43E44E45E46E47E48E49EEquality of mixed partial derivatives Verify that fxy = fyx for the following functions. 44.f(x,y)=xy51EEquality of mixed partial derivatives Verify that fxy = fyx for the following functions. 52. f(x, y) = esin xy53E54E55E56E57E58E59E60E61E62E63E64E65E66E67E68EGas law calculations Consider the Ideal Gas Law PV = kT, where k 0 is a constant. Solve this equation for V in terms of P and T. a.Determine the rate of change of the volume with respect to the pressure at constant temperature. Interpret the result. b.Determine the rate of change of the volume with respect to the temperature at constant pressure. Interpret the result. c.Assuming k = 1, draw several level curves of the volume function and interpret the results as in Example 6.Body mass index The body mass index (BMI) for an adult human is given by the function B = w/h2, where w is the weight measured in kilograms and h is the height measured in meters. (The BMI for units of pounds and inches is B = 703 w/h2.) a.Find the rate of change of the BMI with respect to weight at a constant height. b.For fixed h, is the BMI an increasing or decreasing function of w? Explain. c.Find the rate of change of the BMI with respect to height at a constant weight. d.For fixed w, is the BMI an increasing or decreasing function of h? Explain.Resistors in parallel Two resistors in an electrical circuit with resistance R1 and R2 wired in parallel with a constant voltage give an effective resistance of R, where 1R=1R1+1R2. a.Find RR1 and RR2 by solving for R and differentiating. b.Find RR1 and RR2 by differentiating implicitly. c.Describe how an increase in R1 with R2 constant affects R. d.Describe how a decrease in R2 with R1 constant affects R.Spherical caps The volume of the cap of a sphere of radius r and thickness h is V=3h2(3rh), for 0 h 2r. a.Compute the partial derivatives Vh and Vr. b.For a sphere of any radius, is the rate of change of volume with respect to r greater when h = 0.2r or when h = 0.8r? c.For a sphere of any radius, for what value of h is the rate of change of volume with respect to r equal to 1? d.For a fixed radius r, for what value of h (0 h 2r) is the rate of change of volume with respect to h the greatest?Heat equation The flow of hear along a thin conducting bar is governed by the one-dimensional heal equation (with analogs for thin plates in two dimensions and for solids in three dimensions) ut=k2ux2, where u is a measure of the temperature at a location x on the bar at time t and the positive constant k is related to the conductivity of the material. Show that the following functions satisfy the heal equation with k = 1. 85.u(x, t) = 4e4t cos 2xHeat equation The flow of hear along a thin conducting bar is governed by the one-dimensional heal equation (with analogs for thin plates in two dimensions and for solids in three dimensions) ut=k2ux2, where u is a measure of the temperature at a location x on the bar at time t and the positive constant k is related to the conductivity of the material. Show that the following functions satisfy the heal equation with k = 1. 84.u(x, t) = 10et sin xHeat equation The flow of hear along a thin conducting bar is governed by the one-dimensional heal equation (with analogs for thin plates in two dimensions and for solids in three dimensions) ut=k2ux2, where u is a measure of the temperature at a location x on the bar at time t and the positive constant k is related to the conductivity of the material. Show that the following functions satisfy the heal equation with k = 1. 87.u(x,t)=Aea2t cos ax, for any real numbers a and A76ENondifferentiability? Consider the following functions f. a.Is f continuous at (0, 0)? b.Is f differentiable at (0, 0)? c.If possible, evaluate fx(0, 0) and fy(0, 0). d.Determine whether fx and fy are continuous at (0, 0). e.Explain why Theorems 12.5 and 12.6 are consistent with the results in parts (a)(d). 57.f(x,y)={xyx2+y2if(x,y)(0,0)0if(x,y)=(0,0)Nondifferentiability? Consider the following functions f. a.Is f continuous at (0, 0)? b.Is f differentiable at (0, 0)? c.If possible, evaluate fx(0, 0) and fy(0, 0). d.Determine whether fx and fy are continuous at (0, 0). e.Explain why Theorems 12.5 and 12.6 are consistent with the results in parts (a)(d). 58.f(x,y)={2xyx2+y4if(x,y)(0,0)0if(x,y)=(0,0)79E80E81E82EElectric potential function The electric potential in the xy-plane associated with two positive charges, one at (0, 1) with twice the magnitude as the charge at (0, 1), is (x,y)=2x2+(y1)2+1x2+(y+1)2. a.Compute x and y. b.Describe how x and y behave as x, y . c.Evaluate x(0, y), for all y 1. Interpret this result. d.Evaluate y(x, 0), for all x. Interpret this result.84E85EWave on a string Imagine a string that is fixed at both ends (for example, a guitar string). When plucked, the string forms a standing wave. The displacement u of the string varies with position x and with time t. Suppose it is given by u = f(x, t) = 2 sin (x) sin (t/2), for 0 x 1 and t 0 (see figure). At a fixed point in time, the string forms a wave on [0, 1]. Alternatively, if you focus on a point on the string (fix a value of x), that point oscillates up and down in time. a.What is the period of the motion in time? b.Find the rate of change of the displacement with respect to time at a constant position (which is the vertical velocity of a point on the string). c.At a fixed time, what point on the string is moving fastest? d.At a fixed position on the string, when is the string moving fastest? e.Find the rate of change of the displacement with respect to position at a constant time (which is the slope of the string). f.At a fixed time, where is the slope of the string greatest?Wave equation Traveling waves (for example, water waves or electromagnetic waves) exhibit periodic motion in both time and position. In one dimension, some types of wave motion are governed by the one-dimensional wave equation 2ut2=c22ux2, where u(x, t) is the height or displacement of the wave surface at position x and time t, and c is the constant speed of the wave. Show that the following functions are solutions of the wave equation. 77.u(x, t) = cos (2(x + ct))Wave equation Traveling waves (for example, water waves or electromagnetic waves) exhibit periodic motion in both time and position. In one dimension, some types of wave motion are governed by the one-dimensional wave equation 2ut2=c22ux2, where u(x, t) is the height or displacement of the wave surface at position x and time t, and c is the constant speed of the wave. Show that the following functions are solutions of the wave equation. 78.u(x, t) = 5 cos (2(x + ct)) + 3 sin (x ct)Wave equation Traveling waves (for example, water waves or electromagnetic waves) exhibit periodic motion in both time and position. In one dimension, some types of wave motion are governed by the one-dimensional wave equation 2ut2=c22ux2, where u(x, t) is the height or displacement of the wave surface at position x and time t, and c is the constant speed of the wave. Show that the following functions are solutions of the wave equation. 79.u(x, t) = A f(x + ct) + B g(x ct), where A and B are constants and f and g are twice differentiable functions of one variableLaplaces equation A classical equation of mathematics is Laplaces equation, which arises in both theory and applications. It governs ideal fluid flow, electrostatic potentials, and the steady-state distribution of heat in a conducting medium. In two dimensions, Laplaces equation is 2ux2+2uy2=0. Show that the following functions are harmonic; that is, they satisfy Laplaces equation. 80.u(x, y) = ex sin yLaplaces equation A classical equation of mathematics is Laplaces equation, which arises in both theory and applications. It governs ideal fluid flow, electrostatic potentials, and the steady-state distribution of heat in a conducting medium. In two dimensions, Laplaces equation is 2ux2+2uy2=0. Show that the following functions are harmonic; that is, they satisfy Laplaces equation. 81.u(x, y) = x(x2 3y2)Laplaces equation A classical equation of mathematics is Laplaces equation, which arises in both theory and applications. It governs ideal fluid flow, electrostatic potentials, and the steady-state distribution of heat in a conducting medium. In two dimensions, Laplaces equation is 2ux2+2uy2=0. Show that the following functions are harmonic; that is, they satisfy Laplaces equation. 82.u(x, y) = eax cos ay, for any real number aLaplaces equation A classical equation of mathematics is Laplaces equation, which arises in both theory and applications. It governs ideal fluid flow, electrostatic potentials, and the steady-state distribution of heat in a conducting medium. In two dimensions, Laplaces equation is 2ux2+2uy2=0. Show that the following functions are harmonic; that is, they satisfy Laplaces equation. 83.u(x,y)=tan1(yx1)tan1(yx+1)94EDifferentiability Use the definition of differentiability to prove that the following functions are differentiable at (0, 0). You must produce functions 1 and 2 with the required properties. 89.f(x, y) = xyNondifferentiability? Consider the following functions f. a.Is f continuous at (0, 0)? b.Is f differentiable at (0, 0)? c.If possible, evaluate fx(0, 0) and fy (0, 0). d.Determine whether fx and fy are continuous at (0, 0). e.Explain why Theorems 12.5 and 12.6 are consistent with the results in parts (a)(d). 90.f(x, y) = 1 |xy|