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All Textbook Solutions for Calculus: Early Transcendentals (3rd Edition)
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). 91.f(x,y)=|xy|98EDerivatives of an integral Let h be continuous for all real numbers. Find fx and fy when f(x,y)=1xyh(s)ds.Explain why Theorem 15.7 reduces to the Chain Rule for a function of one variable in the case that z = f(x) and x = g(t).Suppose w = f(x, y, z), where x = g(s, t), y = h(s, t), and z = p(s, t). Extend Theorem 15.8 to write a formula for w/t.If Q is a function of w, x, y, and z, each of which is a function of r, s, and t, how many dependent variables, intermediate variables, and independent variables are there?Use the method of Example 5 to find dy/dx when F(x, y) = x2+xyy37=0.Compare your solution to Example 3 in Section 3.8. Which method is easier?Suppose z = f(x, y), where x and y are functions of t. How many dependent, intermediate, and independent variables are there?Let z be a function of x and y, while x and y are functions of t. Explain how to find dzdt.Suppose w is a function of x, y and z, which are each functions of t. Explain how to find dwdt.Let z = f(x, y), x = g(s, t), and y = h(s, t). Explain how to find z/t.Given that w = F(x, y, z), and x, y, and z are functions of r and s, sketch a Chain Rule tree diagram with branches labeled with the appropriate derivatives.Suppose F(x, y) = 0 and y is a differentiable function of x. Explain how to find dy/dx.Evaluate dz/dt, where z = x2+y3, x = t2 and y = t, using Theorem 15.7 Check your work by writing z as a function of t and evaluating dz/dt.8EChain Rule with one independent variable Use Theorem 15.7 to find the following derivatives. 9. dz/dt, where z = x sin y, x = t2, and y = 4t3Chain Rule with one independent variable Use Theorem 15.7 to find the following derivatives. 10. dz/dt, where z = x2y xy3, x = t2, and y = t2Chain Rule with one independent variable Use Theorem 15.7 to find the following derivatives. 11. dw/dt, where w = cos 2x sin 3y, x = t/2, and y = t4Chain Rule with one independent variable Use Theorem 15.7 to find the following derivatives. 12. dz/dt, where z = r2+s2, r = cos 2t, and s = sin 2tChain Rule with one independent variable Use Theorem 15.7 to find the following derivatives. 13. dz/dt, where z = (x + 2y)10, x = sin2 t, y = (3t + 4)5Chain Rule with one independent variable Use Theorem 15.7 to find the following derivatives. 14. dzdt, where z =x20y20, x = tan1t, y = ln(t2 +1)Chain Rule with one independent variable Use Theorem 15.7 to find the following derivatives. 15. dw/dt, where w = xy sin z, x = t2, y = 4t3, and z = t + 1Chain Rule with one independent variable Use Theorem 15.7 to find the following derivatives. 16. dQ/dt, where Q = x2+y2+z2, x = sin t, y = cos t, and z = cos tChain Rule with one independent variable Use Theorem 15.7 to find the following derivatives. 17. dV/dt, where V = xyz, x = et, y = 2t + 3, and z = sin tChain Rule with one independent variable Use Theorem 15.7 to find the following derivatives. 18. dUdt, where U=xy2z8, x = et, y = sin 3t, and z = 4t + 1Chain Rule with several independent variables Find the following derivatives. 19.zs and zr, where z = x2 sin y, x = s t, and y = t2Chain Rule with several independent variables Find the following derivatives. 20.zs and zt, where z = sin (2x + y), x = s2 t, and y = s2 + t2Chain Rule with several independent variables Find the following derivatives. 21.zs and zt, where z = xy x2y, x = s + t, and y = s tChain Rule with several independent variables Find the following derivatives. 22.zs and zt, where z = sin x cos 2y, x = s + t, and y = s tChain Rule with several independent variables Find the following derivatives. 23.zs and zt, where z = ex + y, x = st, and y = s + t24EChain Rule with several independent variables Find the following derivatives. 25.ws, and wt where w=xzy+z, x = s + t, y = st, and z = s t26EChanging cylinder The volume of a right circular cylinder with radius r and height h is V = r2h. a.Assume that r and h are functions of t. Find V(t) b.Suppose that r = et and h = e2t, for t 0. Use part (a) to find V(t). c.Does the volume of the cylinder in part (b) increase or decrease as t increases?Changing pyramid The volume of a pyramid with a square base x units on a side and a height of h is V=13x2h. a.Assume that x and h are functions of t. Find V(t). b.Suppose that x = t/(t + 1) and h = 1/(t + 1),for t 0. Use part (a) to find V(t). c.Does the volume of the pyramid in part (b) increase or decrease as t increases?Derivative practice two ways Find the indicated derivative in two ways: a.Replace x and y to write z as a function of t and differentiate. b.Use the Chain Rule. 41.z(t), where z=1x+1y,x=t2+2t, and y = t3 2Derivative practice two ways Find the indicated derivative in two ways: a.Replace x and y to write z as a function of t and differentiate. b.Use the Chain Rule. 40.z(t), where z = ln (x +y), x = tet, and y = etMaking trees Use a tree diagram to write the required Chain Rule formula. 27.w is a function of z, where z, is a function of x and y, each of which is a function of t. Find dw/dt.32E33E34EImplicit differentiation Given the following equations, evaluate dy/dx. Assume that each equation implicitly defines y as a differentiable function of x 31.x2 2y2 1 = 0Implicit differentiation Given the following equations, evaluate dy/dx. Assume that each equation implicitly defines y as a differentiable function of x 32.x3 + 3xy2 y5 = 0Implicit differentiation Given the following equations, evaluate dy/dx. Assume that each equation implicitly defines y as a differentiable function of x. 33.2 sin xy = 1Implicit differentiation Given the following equations, evaluate dy/dx. Assume that each equation implicitly defines y as a differentiable function of x. 34.yexy 2 = 0Implicit differentiation Given the following equations, evaluate dy/dx. Assume that each equation implicitly defines y as a differentiable function of x. 35.x2+2xy+y4=3Implicit differentiation Given the following equations, evaluate dy/dx. Assume that each equation implicitly defines y as a differentiable function of x 36.y ln (x2 + y2 + 4) = 341E42E43E44E45E46E47E48E49EDerivative practice Find the indicated derivative for the following functions. 42.z/p, where z = x/y, x = p + q, and y = p qDerivative practice Find the indicated derivative for the following functions. 43. dw/dt,, where w = xyz, x = 2t4, = 3t1, and z = 4t3Derivative practice Find the indicated derivative for the following functions. 44.w/x, where w = cos z cos x cos y + sin x sin y and z = x + yDerivative practice Find the indicated derivative for the following functions. 45.zx,where1x+1y+1z=154EChange on a line Suppose w=(x,y,z) and is the line r(t)=at,bt,ct, for t a. Find on w(t) (in terms of a, b, c, wx. wy, and wz). b. Apply part (a) to find w(t) when f(x, y, z) = xyz. c. Apply part (a) to find w(t) when f(x,y, z) = x2+y2+z2. d. For a general function w = f(x, y, z), find w(t).56EImplicit differentiation with three variables Use the result of Exercise 56 to evaluate z/x andz/y for the following relations. 57. xy + xz + yz = 358E59EMore than one way Let exyz = 2. Find zx and zy in three ways (and check for agreement). a. Use the result of Exercise 56. b. Take logarithms of both sides and differentiate xyz = ln 2. c. Solve for z and differentiate z = ln2xyWalking on a surface Consider the following surfaces specified in the form z = f(x, y) and the oriented curve C in the xy-plane. a. In each case, find z(t). b. Imagine that you are walking on the surface directly above the curve C in the direction of positive orientation. Find the values of t for which you are walking uphill (that is, z is increasing). 53.z=x2+4y2+1,C:x=cost,y=sint;0t2Walking on a surface Consider the following surfaces specified in the form z = f(x, y) and the oriented curve C in the xy-plane. a. In each case, find z (t). b. Imagine that you are walking on the surface directly above the curve C in the direction of positive orientation. Find the values of t for which you are walking uphill (that is, z is increasing). 54.z=4x2y2+1,C:x=cost,y=sint;0t2Walking on a surface Consider the following surfaces specified in the form z = f(x, y) and the oriented curve C in the xy-plane. a. In each case, find z (t). b. Imagine that you are walking on the surface directly above the curve C in the direction of positive orientation. Find the values of t for which you are walking uphill (that is, z is increasing). 55.z=1x2y2,C:x=et,y=ett12ln2Walking on a surface Consider the following surfaces specified in the form z = f(x, y) and the oriented curve C in the xy-plane. a. In each case, find z (t). b. Imagine that you are walking on the surface directly above the curve C in the direction of positive orientation. Find the values of t for which you are walking uphill (that is, z is increasing). 56.z=2x2+y2+1,C:x=1+cost,y=sint;0t2Conservation of energy A projectile with mass m is launched into the air on a parabolic trajectory. For t 0, its horizontal and vertical coordinates are x(t) = u0 t andy(t)=12gt2+v0t, respectively. where u0 is the initial horizontal velocity, v0 is the initial vertical velocity, and g is the acceleration due to gravity. Recalling that u(t)=x(t) and v(t)=y(t) are the components of the velocity, the energy of the projectile (kinetic plus potential) is E(t)=12m(u2+v2)+mgy. Use the Chain Rule to compute E(t)and show that E(t)=0, for all t 0. Interpret the result.Utility functions in economics Economists use utility functions to describe consumers relative preference for two or more commodities (for example, vanilla vs. chocolate ice cream or leisure time vs. material goods). The Cobb-Douglas family of utility functions has the form U(x,y)=xay1a, where x and y are the amounts of two commodities and 0 a l is a parameter. Level curves on which the utility function is constant are called indifference curves; the utility is the same for all combinations of x and y along an indifference curve (see figure). a. The marginal utilities of the commodities x and y are defined to be U/x and U/y, respectively. Compute the marginal utilities for the utility function U(x,y)=xay1a. b. The marginal rate of substitution (MRS) is the slope of the indifference curve at the point (x, y). Use the Chain Rule to show that for U(x,y)=xay1a,the MRS is a1ayx. c. Find the MRS for the utility function U(x,y)=x0.4y0.6at (x,y) = (8,12).Constant volume tori The volume of a solid torus is given by V=(2/4)(R+r)(Rr)2, where r and R are the inner and outer radii and R r (see figure). a. If R and r increase at the same rate, does the volume of the torus increase, decrease, or remain constant? b. If R and r decrease at the same rate, does the volume of the torus increase, decrease, or remain constant?Body surface area One of several empirical formulas that relates the surface area S of a human body to the height h and weight w of the body is the Mosteller formula S(h,w)=160hw, where h is measured in centimeters, w is measured in kilograms, and S is measured in square meters. Suppose that h and w are functions of t. a. Find S(t) b. Show that the condition that the surface area remains constant as h and w change is wh(t)+hw(t)=0. c. Show that part (b) implies that for constant surface area, h and w must be inversely related; that is. h = C/w, where C is a constant.The Ideal Gas Law The pressure, temperature, and volume of an ideal gas are related by PV = kT, where k 0 is a constant. Any two of the variables may be considered independent, which determines the dependent variable. a. Use implicit differentiation to compute the partial derivatives PV,TP,and VT. b. Show that PVTPVT = 1 (See Exercise 75 for a generalization.)70E71EChange of coordinates Recall that Cartesian and polar coordinates are related through the transformation equations {x=rcosy=rcosor{r2=x2+y2tan=y/x a. Evaluate the partial derivatives xr, yr, x , and y b. Evaluate the partial derivatives rx, ry, x, and y. c. For a function z = f(x, y), find zr and z where x and y are expressed in terms of r and . d. For a function z = g(r, ), find zx and zy, where r and are expressed in terms of x and y. e. Show that (zx)2+(zy)2=(zr)2+1r2(z)2.Change of coordinates continued An important derivative operation in many applications is called the Laplacian; in Cartesian coordinates, for z = f(x, y), the Laplacian is zxx + zyy Determine the Laplacian in polar coordinates using the following steps. a. Begin with z = g(r, ) and write zx and zy in terms of polar coordinates (see Exercise 64). b. Use the Chain Rule to find zxx=x(zx). There should be two major terms, which, when expanded and simplified, result in five terms. c. Use the Chain Rule to find zyy=x(zy). There should be two major terms, which, when expanded and simplified, result in five terms. d. Combine parts (b) and (c) to show that zxx+zyy=zrr1rzr+1r2z.75E76E77EExplain Why, when u = 1, 0 in the definition of the directional derivative, the result is fx(a, b), and when u = 0, 1, the result is fy(a, b).In the parametric description x = a + su1 and y = b + su2, where u = u1, u2 is a unit vector, show that any positive change s ins produces a line segment of length s.In Example 1, evaluate Du f(3, 2) and Dv f(3, 2). Example 1 Computing directional derivatives Consider the paraboloid z = f(x, y) = 14(x2+2y2)+2. Let P0 be the point (3, 2) and consider the unit vectors u = 12,12 and v = 12,32 a. Find the directional derivative of f at P0 in the directions of u and v.Draw a circle in the xy-plane centered at the origin and regard it is as a level curve of the surface z = x2 + y2. At the point (a, a) of the level curve in the xy-plane, the slope of the tangent line is −1. Show that the gradient at (a, a) is orthogonal to the tangent line.
5QC6QC1EHow do you compute the gradient of the functions f(x, y) and f (x, y, z)?3E4EGiven a function f, explain the relationship between the gradient and the level curves of f.The level curves of the surface z=x2+y2 are circles in the xy-plane centered at the origin. Without computing the gradient, what is the direction of the gradient at (1, 1) and (1, 1) (determined up to a scalar multiple)?Suppose f is differentiable at (3, 4), f(3, 4) = 3,1, and u = 32,12 compute Du f(3, 4)Suppose f is differentiable at (9, 9), f(9, 9) = 3,1,and w = 1,1. compute the directional derivative of f at (9, 9) in the direction of the vector w.Suppose f is differentiable at (3, 4). Assume u, v, and w are unit vectors, v = u, w f(3, 4) = 0, and Du f(3, 4) = 7. Find Dv f(3, 4)and Dw f(3, 4)Suppose f is differentiable at (1, 2) and ∇ f(1, 2) = ⟨3, 4⟩. Find the slope of the line tangent to the level curve of f at (1, 2).
Directional derivatives Consider the function f(x.y) = 8x2/2 y2. whose graph is a paraboloid (see figure). a.Fill in the table with the values of the directional derivative at the points (a, b) in the directions given by the unit vectors u. v. and w. (a,b) = (2,0) (a, b) = (0,2) (a, b) = (1,1) u=22,22 v=22,22 w=22,22 b.Interpret each of the directional derivatives computed in pan (a) at the point (2,0).Directional derivatives Consider the function f(x,y)=2x2+y2, whose graph is a paraboloid (see figure). a.Fill in the table with the values of the directional derivative at the points (a, b) in the directions given by the unit vectors u, v, and w. (a,b) = (1,0) (a,b) = (1,1) (a,b) = (1,2) u=1,0 v=22,22 w=0,1 b.Interpret each of the directional derivatives computed in part (a) at the point (1, 0).Computing gradients Compute the gradient of the following functions and evaluate it at the given point P 9.f(x,y)=2+3x25y2;P(2,1)Computing gradients Compute the gradient of the following functions and evaluate it at the given point P 10.f(x,y)=4x22xy+y2;P(1,5)Computing gradients Compute the gradient of the following functions and evaluate it at the given point P 11.g(x,y)=x24x2y8xy2;P(1,2)Computing gradients Compute the gradient of the following functions and evaluate it at the given point P 12.p(x,y)=124x2y2;P(1,1)Computing gradients Compute the gradient of the following functions and evaluate it at the given point P 13.f(x,y)=xe2xy;P(1,0)Computing gradients Compute the gradient of the following functions and evaluate it at the given point P 14.f(x,y)=sin(3x+2y);P(,3/2)Computing gradients Compute the gradient of the following functions and evaluate it at the given point P 15.F(x,y)=ex22y2;P(1,2)Computing gradients Compute the gradient of the following functions and evaluate it at the given point P 16h(x,y)=ln(1+x2+2y2);P(2,3)Computing directional derivatives with the gradient Compute the directional derivative of the following functions at the given point P in the direction of the given vector. Be sure to use a unit vector for the direction vector. 17.f(x,y)=x2y2;P(1,3);35,45Computing directional derivatives with the gradient Compute the directional derivative of the following functions at the given point P in the direction of the given vector. Be sure to use a unit vector for the direction vector. 18f(x,y)=3x2y3;P(3,2);513,121323EComputing directional derivatives with the gradient Compute the directional derivative of the following functions at the given point P in the direction of the given vector. Be sure to use a unit vector for the direction vector. 20g(x,y)=sin(2xy);P(1,1);513,1213Computing directional derivatives with the gradient Compute the directional derivative of the following functions at the given point P in the direction of the given vector. Be sure to use a unit vector for the direction vector. 21f(x,y)=4x22y;P(2,2);15,2526EComputing directional derivatives with the gradient Compute the directional derivative of the following functions at the given point P in the direction of the given vector. Be sure to use a unit vector for the direction vector. 23f(x,y)=3x2+2y+5;P(1,2);3,4Computing directional derivatives with the gradient Compute the directional derivative of the following functions at the given point P in the direction of the given vector. Be sure to use a unit vector for the direction vector. 24h(x,y)=exy;P(ln2,ln3);1,1Computing directional derivatives with the gradient Compute the directional derivative of the following functions at the given point P in the direction of the given vector. Be sure to use a unit vector for the direction vector. 23g(x,y)=ln(4+x2+y2);P(1,2);2,1Computing directional derivatives with the gradient Compute the directional derivative of the following functions at the given point P in the direction of the given vector. Be sure to use a unit vector for the direction vector. 26f(x,y)=x/(xy);P(4,1);1,2Direction of steepest ascent and descent Consider the following functions and points P a. Find the unit vectors that give the direction of steepest ascent and steepest descent at P. b. Find a vector that points in a direction of no change in the function at P. 27f(x,y)=x24y29;P(1,2)Direction of steepest ascent and descent Consider the following functions and points P a. Find the unit vectors that give the direction of steepest ascent and steepest descent at P. b. Find a vector that points in a direction of no change in the function at P. 28f(x,y)=x2+4xyy2;P(2,1)Direction of steepest ascent and descent Consider the following functions and points P a. Find the unit vectors that give the direction of steepest ascent and steepest descent at P. b. Find a vector that points in a direction of no change in the function at P. 29f(x,y)=x4x2y+y2+6;P(1,1)Direction of steepest ascent and descent Consider the following functions and points P a. Find the unit vectors that give the direction of steepest ascent and steepest descent at P. b. Find a vector that points in a direction of no change in the function at P. 30p(x,y)=20+x2+2xyy2;P(1,2)Direction of steepest ascent and descent Consider the following functions and points P a. Find the unit vectors that give the direction of steepest ascent and steepest descent at P. b. Find a vector that points in a direction of no change in the function at P. 31F(x,y)=ex2/2y2/2,P(1,1)Direction of steepest ascent and descent Consider the following functions and points P a.Find the unit vectors lhat give the direction of steepest ascent and steepest descent at P. b.Find a vector that points in a direction of no change in the function at P. 32f(x,y)=2sin(2x3y);P(0,)Interpreting directional derivatives A function f and a point P are given. Let correspond to the direction of the directional derivative. a. Find the gradient and evaluate it at P. b. Find the angles (with respect to the positive x-axis) associated with the directions of maximum increase, maximum decrease, and zero change. c. Write the directional derivative at P as a function of ; call this function g. d. Find the value of that maximizes g() and find the maximum value. e. Verify that the value of that maximizes g corresponds to the direction of the gradient. Verify that the maximum value of g equals the magnitude of the gradient. 33.f(x,y)=102x23y2;P(3,2)Interpreting directional derivatives A function f and a point P are given. Let correspond to the direction of the directional derivative. a. Find the gradient and evaluate it at P. b. Find the angles (with respect to the positive x-axis) associated with the directions of maximum increase, maximum decrease, and zero change. c. Write the directional derivative at P as a function of ; call this function g. d. Find the value of that maximizes g() and find the maximum value. e. Verify that the value of that maximizes g corresponds to the direction of the gradient. Verify that the maximum value of g equals the magnitude of the gradient. 33.f(x,y)=8+x2+3y2;P(3,1)Interpreting directional derivatives A function f and a point P are given. Let correspond to the direction of the directional derivative. a. Find the gradient and evaluate it at P. b. Find the angles (with respect to the positive x-axis) associated with the directions of maximum increase, maximum decrease, and zero change. c. Write the directional derivative at P as a function of ; call this function g. d. Find the value of that maximizes g() and find the maximum value. e. Verify that the value of that maximizes g corresponds to the direction of the gradient. Verify that the maximum value of g equals the magnitude of the gradient. 35.f(x,y)=2+x2+y2;P(3,1)Interpreting directional derivatives A function f and a point P are given. Let correspond to the direction of the directional derivative. a. Find the gradient and evaluate it at P. b. Find the angles (with respect to the positive x-axis) associated with the directions of maximum increase, maximum decrease, and zero change. c. Write the directional derivative at P as a function of ; call this function g. d. Find the value of that maximizes g() and find the maximum value. e. Verify that the value of that maximizes g corresponds to the direction of the gradient. Verify that the maximum value of g equals the magnitude of the gradient. 36.f(x,y)=12x2y2;P(1,1/3)Interpreting directional derivatives A function f and a point P are given. Let correspond to the direction of the directional derivative. a. Find the gradient and evaluate it at P. b. Find the angles (with respect to the positive x-axis) associated with the directions of maximum increase, maximum decrease, and zero change. c. Write the directional derivative at P as a function of ; call this function g. d. Find the value of that maximizes g() and find the maximum value. e. Verify that the value of that maximizes g corresponds to the direction of the gradient. Verify that the maximum value of g equals the magnitude of the gradient. 37.f(x,y)=ex22y2;P(1,0)Interpreting directional derivatives A function f and a point P are given. Let correspond to the direction of the directional derivative. a. Find the gradient and evaluate it at P. b. Find the angles (with respect to the positive x-axis) associated with the directions of maximum increase, maximum decrease, and zero change. c. Write the directional derivative at P as a function of ; call this function g. d. Find the value of that maximizes g() and find the maximum value. e. Verify that the value of that maximizes g corresponds to the direction of the gradient. Verify that the maximum value of g equals the magnitude of the gradient. 38.f(x,y)=ln(1+2x2+3y2);P(343)Directions of change Consider the following functions f and points P. Sketch the xy-plane showing P and the level curve through P. Indicate (as in Figure 15.52) the directions of maximum increase, maximum decrease, and no change for f. 43. f(x, y) = 8 + 4x2 + 2y2; P(2, 4)44E45E43-46. Directions of change Consider the following functions f and points P. Sketch the xy-plane showing P and the level curve through P. Indicate (as in Figure 15.52) the directions of maximum increase, maximum decrease, and no change for f.
46. f(x, y) = tan (2x + 2y); P(π/16, π/16)
Level curves Consider the paraboloid f(x, y) = 16 x2/4 y,2/16 and the point P on the given level curve of f. Compute the slope of the line tangent to the level curve at P and verify that the tangent line is orthogonal to the gradient at that point. 43f(x,y)=0;P(0,16)Level curves Consider the paraboloid f(x, y) = 16 x2/4 y,2/16 and the point P on the given level curve of f. Compute the slope of the line tangent to the level curve at P and verify that the tangent line is orthogonal to the gradient at that point. 44f(x,y)=0;P(8,0)Level curves Consider the paraboloid f(x, y) = 16 x2/4 y,2/16 and the point P on the given level curve of f. Compute the slope of the line tangent to the level curve at P and verify that the tangent line is orthogonal to the gradient at that point. 45f(x,y)=12;P(4,0)Level curves Consider the paraboloid f(x, y) = 16 x2/4 y,2/16 and the point P on the given level curve of f. Compute the slope of the line tangent to the level curve at P and verify that the tangent line is orthogonal to the gradient at that point. 46f(x,y)=12;P(2,3,4)Level curves Consider the upper half of the ellipsoid f(x,y)=1x24y216 and the point P on the given level curve of f. Compute the slope of the line tangent to the level curve at P and verify that the tangent line is orthogonal to the gradient at that point. 47.f(x,y)=3/2;P(1/2,3)Level curves Consider the upper half of the ellipsoid f(x,y)=1x24y216 and the point P on the given level curve of f. Compute the slope of the line tangent to the level curve at P and verify that the tangent line is orthogonal to the gradient at that point. 48.f(x,y)=1/2;P(0,8)Level curves Consider the upper half of the ellipsoid f(x,y)=1x24y216 and the point P on the given level curve of f. Compute the slope of the line tangent to the level curve at P and verify that the tangent line is orthogonal to the gradient at that point. 48.f(x,y)=1/2;P(2,0)54EPath of steepest descent Consider each of the following surfaces and the point P on the surface. a. Find the gradient of f. b. Let C be the path of steepest descent on the surface beginning at P and let C be the projection of C on the xy-plane. Find an equation of C in the xy-plane. 51f(x,y)=4+x(aplane);P(4,4,8)Path of steepest descent Consider each of the following surfaces and the point P on the surface. a. Find the gradient of f. b. Let C be the path of steepest descent on the surface beginning at P and let C be the projection of C on the xy-plane. Find an equation of C in the xy-plane. 52f(x,y)=y+x(aplane);P(2,2,4)Path of steepest descent Consider each of the following surfaces and the point P on the surface. a. Find the gradient of f. b. Let C be the path of steepest descent on the surface beginning at P and let C be the projection of C on the xy-plane. Find an equation of C in the xy-plane. 53f(x,y)=4x22y2(aparaboloid);P(1,1,1)Path of steepest descent Consider each of the following surfaces and the point P on the surface. a. Find the gradient of f. b. Let C be the path of steepest descent on the surface beginning at P and let C be the projection of C on the xy-plane. Find an equation of C in the xy-plane. 54f(x,y)=y+x1;P(1,2,3)Gradients in three dimensions Consider the following functions f, points P, and unit vectors u. a. Compute the gradient of f and evaluate it at P b. Find the unit vector in the direction of maximum increase of f at P. c. Find the rate of change of the function in the direction of maximum increase at P. d. Find the directional derivative at P in the direction of the given vector. 55.f(x,y,z)=x2+2y2+4z2+10;P(1,0,4);12,0,12Gradients in three dimensions Consider the following functions f, points P, and unit vectors u. a. Compute the gradient of f and evaluate it at P b. Find the unit vector in the direction of maximum increase of f at P. c. Find the rate of change of the function in the direction of maximum increase at P. d. Find the directional derivative at P in the direction of the given vector. 56.f(x,y,z)=4x2+3y2+z22;P(0,2,1);0,12,12Gradients in three dimensions Consider the following functions f, points P, and unit vectors u. a. Compute the gradient of f and evaluate it at P b. Find the unit vector in the direction of maximum increase of f at P. c. Find the rate of change of the function in the direction of maximum increase at P. d. Find the directional derivative at P in the direction of the given vector. 56.f(x,y,z)=4x2+3y2+z2z;P(0,2,1);0,12,12Gradients in three dimensions Consider the following functions f, points P, and unit vectors u. a. Compute the gradient of f and evaluate it at P b. Find the unit vector in the direction of maximum increase of f at P. c. Find the rate of change of the function in the direction of maximum increase at P. d. Find the directional derivative at P in the direction of the given vector. 58.f(x,y,z)=xy+yz+xz+4;P(2,2,1);0,12,12Gradients in three dimensions Consider the following functions f, points P, and unit vectors u. a. Compute the gradient of f and evaluate it at P b. Find the unit vector in the direction of maximum increase of f at P. c. Find the rate of change of the function in the direction of maximum increase at P. d. Find the directional derivative at P in the direction of the given vector. 59.f(x,y,z)=1+sin(x+2yz);P(6,6,6);132323Gradients in three dimensions Consider the following functions f, points P, and unit vectors u. a. Compute the gradient of f and evaluate it at P b. Find the unit vector in the direction of maximum increase of f at P. c. Find the rate of change of the function in the direction of maximum increase at P. d. Find the directional derivative at P in the direction of the given vector. 60.f(x,y,z)=exyx1;P(0,1,1);232313Gradients in three dimensions Consider the following functions f, points P, and unit vectors u. a. Compute the gradient of f and evaluate it at P b. Find the unit vector in the direction of maximum increase of f at P. c. Find the rate of change of the function in the direction of maximum increase at P. d. Find the directional derivative at P in the direction of the given vector. 61.f(x,y,z)=ln(1+x2+y2+z2);P(1,1,1);232313Gradients in three dimensions Consider the following functions f, points P, and unit vectors u. a. Compute the gradient of f and evaluate it at P b. Find the unit vector in the direction of maximum increase of f at P. c. Find the rate of change of the function in the direction of maximum increase at P. d. Find the directional derivative at P in the direction of the given vector. 62.f(x,y,z)=xzyz;P(3,2,1);132313Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. If f(x,y)=x2+y210, then f(x,y)=2x+2y. b. Because the gradient gives the direction of maximum increase of a function, the gradient is always positive. c. The gradient of f(x,y,z)=1+xyz has four components. d. If f(x,y,z) = 4, then f=0Gradient of a composite function Consider the function F(x,y,z)=exyz. a. Write F as a composite function fg, where f is a function of one variable and g is a function of three variables. b. Relate F to g.Directions of zero change Find the directions in the xy-plane in which the following functions have zero change at the given point. Express the directions in terms of unit vectors. 65.f(x,y)=124x2y2;P(1,2,4)70EDirections of zero change Find the directions in the xy-plane in which the following functions have zero change at the given point. Express the directions in terms of unit vectors. 67.f(x,y)=3+2x2+y2;P(1,2,3)Directions of zero change Find the directions in the xy-plane in which the following functions have zero change at the given point. Express the directions in terms of unit vectors. 68.f(x,y)=e1xy;P(1,0,e)Steepest ascent on a plane Suppose a long sloping hillside is described by the plane z = ax + by + c, where a, b, and c are constants. Find the path in the xy-plane, beginning at (x0, y0), that corresponds to the path of steepest ascent on the hillside.Gradient of a distance function Let (a, b) be a given point in R2 and let d = f(x,y) be the distance between (a, b) and the variable point (x, y). a. Show that the graph of f is a cone. b. Show that the gradient of f at any point other than (a, b) is a unit vector. c. Interpret the direction and magnitude of f.Looking aheadtangent planes Consider the following surfaces f(x, y, z) = 0, which may be regarded as a level surface of the function w = f(x, y z). A point P(a. b, c) on the surface is also given. a. Find the (three-dimensional) gradient off and evaluate it at P b. The set of all vectors orthogonal to the gradient with their tails at P form a plane. Find an equation of that plane (soon to be called the tangent plane). 71.f(x,y,z)=x2+y2+z23=0;P(1,1,1)76ELooking aheadtangent planes Consider the following surfaces f(x, y, z) = 0, which may be regarded as a level surface of the function w = f(x, y z). A point P(a. b, c) on the surface is also given. a. Find the (three-dimensional) gradient off and evaluate it at P b. The set of all vectors orthogonal to the gradient with their tails at P form a plane. Find an equation of that plane (soon to be called the tangent plane). 73.f(x,y,z)=ex+yz1=0;P(1,1,2)78E79E80EPotential functions Potential functions arise frequently in physics and engineering. A potential function has the property that a field of interest (for example, an electric field, a gravitational field, or a velocity field) is the gradient of the potential (or sometimes the negative of the gradient of the potential). (Potential functions are considered in depth in Chapter 17.) 81. Electric potential due to a point charge The electric field due to a point charge of strength Q at the origin has a potential function = kQ/r, where r2 = x2 + y2 + z2 is the square of the distance between a variable point P(x, y, z) and the charge, and k 0 is a physical constant. The electric field is given by E = where is the gradient in three dimensions. a. Show that the three-dimensional electric field due to a point charge is given by E(x,y,z)=kQxr3,yr3,zr3 b. Show that the electric field at a point has a magnitude |E|=kQr2 Explain why this relationship r is called an inverse square law.Potential functions Potential functions arise frequently in physics and engineering. A potential function has the property that a field of interest (for example, an electric field, a gravitational field, or a velocity field) is the gradient of the potential (or sometimes the negative of the gradient of the potential). (Potential functions are considered in depth in Chapter 17.) 82. Gravitational potential The gravitational potential associated with two objects of mass M and m is = GMm/r, where G is the gravitational constant. If one of the objects is at the origin and the other object is at P(x, y, z), then r2= x2 + y2 + z2 is the square of the distance between the objects. The gravitational field at P is given by F = where is the gradient in three dimensions. Show that the force has a magnitude. |F| = GMm/r2.Explain why this relationship is called an inverse square law.83E84ERules for gradients Use the definition of the gradient (in two or three dimensions), assume that f and g are differentiable functions on 2 or 3, and let c be a constant. Prove the following gradient rules. a.Constants Rule: (cf) = cf b.Sum Rule: (f + g) = f + g c.Product Rule: (fg) = (f)g + fg d.Quotient Rule: (fg)=gffgg2 e.Chain Rule: (fg)=f(g)g, where f is a function of one variable86E87E88EUsing gradient rules Use the gradient rules of Exercise 85 to find the gradient of the following functions. 89. f(x,y,z)=25x2y2z2Using gradient rules Use the gradient rules of Exercise 85 to find the gradient of the following functions. 90. f(x, y, z) = (x + y+ z)exyz91EWrite the function z = xy + x y in the form F(x, y, z) = 0.2QC3QC4QCSuppose n is a vector normal to the tangent plane of the surface F(x, y, z) = 0 at a point. How is n related to the gradient of F at that point?Write the explicit function z = xy2 + x2y 10 in the implicit form F(x, y, z) = 0.Write an equation for the plane tangent to the surface F(x, y, z) = 0 at the point (a, b, c)4EExplain how to approximate a function f at a point near (a, b) where the values of f, fx, and fy are known at (a, b).Explain how to approximate the change in a function f when the independent variables change from (a, b) to (a + x, b + y).Write the approximate change formula for a function z = f(x, y) at the point (a, b) in terms of differentials.Write the differential dw for the function w = f(x, y, z).Suppose f(1, 2) = 4, fx(1, 2) = 5, and fy(1, 2) = 3. 9. Find an equation of the plane tangent to the surface z = f(x, y) at the point P0(1, 2, 4).Suppose f(l, 2) = 4, fx(1, 2) = 5, and fy(1, 2) = 3. 10. Find the linear approximation to f at P0(1, 2, 4), and use it to estimate f(1.01, 1.99).Suppose F(0, 2, 1) = 0, Fx(0, 2, 1) = 3, Fy(0, 2, 1) = 1, and Fz = (0, 2, 1) = 6. 11. Find an equation of the plane tangent to the surface F(x, y, z) = 0 at the point P0(0, 2, 1)Suppose F(0, 2, 1) = 0, Fx(0, 2, 1) = 3, Fy(0, 2, 1) = 1, and Fz = (0, 2, 1) = 6. 12. Find the linear approximation to the function w = F(x, y, z) at the point P0(0, 2, 1) and use it to estimate F(0.1, 2, 0.99).Tangent planes for F(x,y,z) = 0 Find an equation of the plane tangent to the following surfaces at the given points (two planes and two equations). 9.x2 + y + z = 3; (1,1, 1) and (2, 0,1)Tangent planes for F(x,y,z) = 0 Find an equation of the plane tangent to the following surfaces at the given points (two planes and two equations). 10.x2 + y3 + z4 = 2; (1, 0, 1) and (1, 0, 1)Tangent planes for F(x,y,z) = 0 Find an equation of the plane tangent to the following surfaces at the given points (two planes and two equations). 11.xy + xz + yz 12 = 0; (2, 2, 2) and (2, 0, 6)Tangent planes for F(x,y,z) = 0 Find an equation of the plane tangent to the following surfaces at the given points (two planes and two equations). 12.x2 + y2 z2 =; (3, 4, 5) and (4,3, 5)Tangent planes for z = f (x, y) Find an equation of the plane tangent to the following surfaces at the given points (two planes and two equations). 17.z = 4 2x2 y2; (2, 2, 8) and (1, 1, 1)Tangent planes for z = f (x, y) Find an equation of the plane tangent to the following surfaces at the given points (two planes and two equations). 18.z=2+2x2+y22;(12,1,3)and (3, 2, 22)Tangent planes for z = f (x, y) Find an equation of the plane tangent to the following surfaces at the given points (two planes and two equations). 19.z = exy; (1, 0, 1) and (0, 1, 1)Tangent planes for z = f (x, y) Find an equation of the plane tangent to the following surfaces at the given points (two planes and two equations). 20.z = sin xy + 2; (1, 0, 2) and (0, 5, 2)Tangent planes for F(x,y,z) = 0 Find an equation of the plane tangent to the following surfaces at the given points (two planes and two equations). 13.xysinz=1;(1,2,/6) and (2,1,5/6)Tangent planes for F(x, y, z) = 0 Find an equation of the plane tangent to the following surfaces at the given points (two planes and two equations). 14.yzexz 8 = 0; (0, 2, 4) and (0, 8, 1)Tangent planes for F(x, y, z) = 0 Find an equation of the plane tangent to the following surfaces at the given points (two planes and two equations). 15.z2x2/16y2/91=0;(4,3,3) and (8,9,14)Tangent planes for F(x, y, z) = 0 Find an equation of the plane tangent to the following surfaces at the given points (two planes and two equations). 16.2x + y2 z2 = 0; (0, 1, 1) and (4, 1, 3)Tangent planes for z = f (x, y) Find an equation of the plane tangent to the following surfaces at the given points (two planes and two equations). 21.z = x2 exy; (2, 2,4) and (1, 1, 1)26ETangent planes for z = f (x, y) Find an equation of the plane tangent to the following surfaces at the given points (two planes and two equations). 23.z=(xy)/(x2+y2);(1,2,15) and (2,1,35)Tangent planes for z = f (x, y) Find an equation of the plane tangent to the following surfaces at the given points (two planes and two equations). 24.z=2cos(xy)+2;(/6,/6,3) and (/3,/3,4)Tangent planes Find an equation of the plane tangent to the following surfaces at the given point. 47.z=tan1xy;(1,1,/4)Tangent planes Find an equation of the plane tangent to the following surfaces at the given point. 46.z = tan1 (x + y); (0, 0, 0)Tangent planes Find an equation of the plane tangent to the following surfaces at the given point. 49.sinxyz=12;(,1,16)Tangent planes Find an equation of the plane tangent to the following surfaces at the given point. 48.(x+z)/(yz)=2;(4,2,0)Linear approximation a.Find the linear approximation to the function f at the given point. b.Use part (a) to estimate the given function value. 25.f(x, y) = xy + x y; (2, 3); estimate f(2.1, 2.99).Linear approximation a.Find the linear approximation to the function f at the given point. b.Use part (a) to estimate the given function value. 26.f(x, y) = 12 4x2 8y2; (1, 4); estimate f(1.05, 0.95).Linear approximation a.Find the linear approximation to the function f at the given point. b.Use part (a) to estimate the given function value. 27.f(x, y) = x2 + 2y2; (3, 1); estimate f(3.1, 1.04)Linear approximation a.Find the linear approximation to the function f at the given point. b.Use part (a) to estimate the given function value. 28.f(x,y)=x2+y2;(3,4); estimate f(3.06, 3.92)Linear Approximation a. Find the linear approximation to the function fat the given point. b. Use part (a) to estimate the given function value. 37. f(x, y, z) ln(1 + x + y +2z); (0, 0, 0); estimate f(0.1, 0.2, 0.2).Linear Approximation a. Find the linear approximation to the function fat the given point. b. Use part (a) to estimate the given function value. 38. f(x,y,z)=x+yxz;(3, 2, 4); estimate f(2.95, 2.05, 4.02).Approximate function change Use differentials to approximate the change in z for the given changes in the independent variables. 31.z = 2x 3y 2xy when (x, y) changes from (1, 4) to (1.1, 3.9)Approximate function change Use differentials to approximate the change in z for the given changes in the independent variables. 32.z = x2 + 3y2 + 2 when (x, y) changes from (1, 2) to (1.05, 1.9)Approximate function change Use differentials to approximate the change in z for the given changes in the independent variables. 33.z = ex+y when (x, y) changes from (0, 0) to (0.1, 0.05)Approximate function change Use differentials to approximate the change in z for the given changes in the independent variables. 34.z = ln (1 + x + y) when (x, y) changes from (0, 0) to (0.1, 0.03)Changes in torus surface area The surface area of a torus with an inner radius r and an outer radius R r is S = 42(R2 r2). a.If r increases and R decreases, does S increase or decrease, or is it impossible to say? b.If r increases and R increases, does S increase or decrease, or is it impossible to say? c.Estimate the change in the surface area of the torus when r changes from r = 3.00 to r = 3.05 and R changes from R = 5.50 to R = 5.65. d.Estimate the change in the surface area of the torus when r changes from r = 3.00 to r = 2.95 and R changes from R = 7.00 to R = 7.04. e.Find the relationship between the changes in r and R that leaves the surface area (approximately) unchanged.Changes in cone volume The volume of a right circular cone with radius r and height h is V=r2h/3. a.Approximate the change in the volume of the cone when the radius changes from r = 6.5 to r = 6.6 and the height changes from h = 4.20 to h = 4.15. b.Approximate the change in the volume of the cone when the radius changes from r = 5.40 to r = 5 37 and the height changes from h = 12.0 to h = 11.96.Area of an ellipse The area of an ellipse with axes of length 2a and 2b is A = ab Approximate the percent change in the area when a increases by 2% and b increases by 1.5%.Volume of a paraboloid The volume of a segment of a circular paraboloid (see figure) with radius r and height h is V=r2h/2. Approximate the percent change in the volume when the radius decreases by 1.5% and the height increases by 2.2%.Differentials with more than two variables Write the differential dw in terms of the differentials of the independent variables. 39.w = f(x, y, z) = xy2 + x2z + yz2Differentials with more than two variables Write the differential dw in terms of the differentials of the independent variables. 40.w = f(x, y, z) = sin (x + y z)Differentials with more than two variables Write the differential dw in terms of the differentials of the independent variables. 41.w=f(u,x,y,z)=(u+x)/(y+z)Differentials with more than two variables Write the differential dw in terms of the differentials of the independent variables. 42.w=f(p,q,r,s)=pq/(rs)Law of Cosines The side lengths of any triangle are related by the Law of Cosines. c2 = a2 + b2 2ab cos . a.Estimate the change in the side length c when a changes from a = 2 to a = 2.03, b changes from b = 4.00 to b = 3.96, and changes from =/3 to =/3+/90. b.If a changes from a = 2 to a = 2.03 and b changes from b = 4.00 to b = 3.96, is the resulting change in c greater in magnitude when =/20 (small angle) or when =9/20 (close to a right angle)?Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a.The planes tangent to the cylinder x2 + z2 = 1 in 3 all have the form ax + bz + c = 0. b.Suppose w=xy/z, for x 0, y 0, and z. 0. A decrease in z with x and y fixed results in an increase in w. c.The gradient F(a, b, c) lies in the plane tangent to the surface F(x, y, z) = 0 at (a, b, c).Horizontal tangent planes Find the points at which the following surfaces have horizontal tangent planes. 52.x2 + 2y2 + z2 2x 2z 2 = 0Horizontal tangent planes Find the points at which the following surfaces have horizontal tangent planes. 51.x2 + y2 z2 2x + 2y + 3 = 0Horizontal tangent planes Find the points at which the following surfaces have horizontal tangent planes. 50.z = sin (x y) in the region 2 x 2, 2 y 2Horizontal tangent planes Find the points at which the following surfaces have horizontal tangent planes. 53.z = cos 2x sin y in the region x , y58ESurface area of a cone A cone with height h and radius r has a lateral surface area (the curved surface only, excluding the base) of S=rr2+h2. a.Estimate the change in the surface area when r increases from r = 2.50 to r = 2.55 and h decreases from h = 0.60 to h = 0.58. b.When r = 100 and h = 200, is the surface area more sensitive to a small change in r or a small change in h? Explain.Line tangent to an intersection curve Consider the paraboloid z = x2 + 3y2 and the plane z = x + y + 4, which intersects the paraboloid in a curve C at (2, 1, 7) (see figure). Find the equation of the line tangent to C at the point (2.1, 7). Proceed as follows. a.Find a vector normal to the plane at (2, 1, 7). b.Find a vector normal to the plane tangent to the paraboloid at (2,1,7). c.Argue that the line tangent to C at (2, 1, 7) is orthogonal to both normal vectors found in parts (a) and (b). Use this fact to find a direction vector for the tangent line. d.Knowing a point on the tangent line and the direction of the tangent line, write an equation of the tangent line in parametric form.Water-level changes A conical tank with radius 0.50 m and height 2.00 m is filled with water (see figure). Water is released from the tank, and the water level drops by 0.05 m (from 2.00 m to 1.95 m). Approximate the change in the volume of water in the tank. (Hint: When the water level drops, both the radius and height of the cone of water change.)63EFloating-point operations In general, real numbers (with infinite decimal expansions) cannot be represented exactly in a computer by floating-point numbers (with finite decimal expansions). Suppose that floating-point numbers on a particular computer carry an error of at most 1016. Estimate the maximum error that is committed in doing the following arithmetic operations. Express the error in absolute and relative (percent) terms. a.f(x, y) = xy b.f(x,y)=x/y c.F(x, y, z) = xz d.F(x,y,z)=(x/y)/zProbability of at least one encounter Suppose that in a large group of people, a fraction 0 r 1 of the people have flu. The probability that in n random encounters you will meet at least one person with flu is P = f(n, r) = 1 (1 r)n. Although n is a positive integer, regard it as a positive real number. a.Compute fr and fn. b.How sensitive is the probability P to the flu rate r? Suppose you meet n = 20 people. Approximately how much does the probability P increase if the flu rate increases from r = 0.1 to r = 0 11 (with n fixed)? c.Approximately how much does the probability P increase if the flu rate increases from r = 0.9 to r = 0.91 with n = 20? d.Interpret the results of parts (b) and (c).Two electrical resistors When two electrical resistors with resistance R1 0 and R2 0 are wired in parallel in a circuit (see figure), the combined resistance R, measured in ohms (), is given by 1R=1R1+1R2 a. Estimate the change in R if R1 increases from 2 to 2.05 and R2 decreases from 3 to 2.95 . b. Is it true that if R1 = R2 and R1 increases by the same small amount as R2 decreases, then R is approximately unchanged? Explain. c. Is it true that if R1 and R2 increase, then R increases? Explain. d. Suppose R1 R2 and R1 increases by the same small amount as R2 decreases. Does R increase or decrease?Three electrical resistors Extending Exercise 66, when three electrical resistors with resistances R1 0, R2 0, and R3 0 are wired in parallel in a circuit (see figure), the combined resistance R, measured in ohms (), is given by 1R1R1+1R2+1R3. Estimate the change in R if R1 increases from 2 to 2.05 , R2 decreases from 3 to 2.95 . and R3 increases from 1.5 to 1.55 .68ELogarithmic differentials Let f be a differentiable function of one or more variables that is positive on its domain. a.Show that d(lnf)=dff. b.Use part (a) to explain the statement that the absolute change in ln f is approximately equal to the relative change in f. c.Let f(x, y) = xy, note that ln f = ln x + ln y, and show that relative changes add: that is, df/f = dx/x + dy/y. d.Let f(x, y) = x/y, note that ln f = ln x = ln y, and show that relative changes subtract; that is df /f = dx/x dy/y. e.Show that in a product of n numbers, f = x1x2xn, the relative change in f is approximately equal to the sum of the relative changes in the variables.The parabola z = x2 + y2 4x + 2y + 5 has a local minimum at (2, 1). Verify the conclusion of Theorem 15.14 for this function.Consider the plane tangent to a surface at a saddle point. In what direction does the normal to the plane point?
Compute the discriminant D(x, y) of f(x, y) = x2y2.Does the linear function f(x, y) = 2x + 3y have an absolute maximum or minimum value on the open unit square {(x, y): 0 x 1, 0 y 1}?Describe the appearance of a smooth surface with a local maximum at a point.Describe the usual appearance of a smooth surface at a saddle point.What are the conditions for a critical point of a function f?If fx (a, b) = fy (a, b) = 0, does it follow the f has a local maximum or local minimum at (a, b)? Explain.Consider the function z = f(x, y). What is the discriminant of f, and how do you compute it?6EWhat is an absolute minimum value of a function f on a set R in 2?What is the procedure for locating absolute maximum and minimum values on a closed bounded domain?Assume the second derivatives of fare continuous throughout the xy-plane and fx(0, 0) = fy(0,0) = 0. Use the given information and the Second Derivative Test to determine whether f has a local minimum, a local maximum, or a saddle point at (0, 0), or state that the test is inconclusive. 9. fxx(0,0) = 5, fyy(0, 0) = 3, and fxy(0,0) = 4Assume the second derivatives of fare continuous throughout the xy-plane and fx(0, 0) = fy(0,0) = 0. Use the given information and the Second Derivative Test to determine whether f has a local minimum, a local maximum, or a saddle point at (0, 0), or state that the test is inconclusive. 10. fxx(0, 0) = 6, fyy(0, 0) = 3, and fxy(0, 0) = 4Assume the second derivatives of fare continuous throughout the xy-plane and fx(0, 0) = fy(0, 0) = 0. Use the given information and the Second Derivative Test to determine whether f has a local minimum, a local maximum, or a saddle point at (0, 0), or state that the test is inconclusive. 11. fxx(0, 0) = 8, fyy(0, 0) = 5, and fxy(0, 0) = 6Assume the second derivatives of fare continuous throughout the xy-plane and fx(0, 0) = fy(0,0) = 0. Use the given information and the Second Derivative Test to determine whether f has a local minimum, a local maximum, or a saddle point at (0, 0), or state that the test is inconclusive. 12. fxx(0, 0) = 9, fyy(0, 0) = 4, and fxy(0, 0) = 6Critical points Find all critical points of the following functions. 12.f(x, y) = 3x3 4y2Critical points Find all critical points of the following functions. 10.f(x, y) = x2 6x + y2 + 8yCritical points Find all critical points of the following functions. 15. f(x, y) = 3x2 + 3y y3Critical points Find all critical points of the following functions. 16. f(x, y) = x3 12x + 6y2Critical points Find all critical points of the following functions. 13.f(x, y) = x4 + y4 16xyCritical points Find all critical points of the following functions. 14.f(x,y)=x3/3y3/3+3xyCritical points Find all critical points of the following functions. 15.f(x, y) = x4 2x2 + y2 4y + 5Critical points Find all critical points of the following functions. 20. f(x, y) = x3 + 6xy 6x + y2 2yCritical points Find all critical points of the following functions. 21. f(x, y) = y3 + 6xy + x2 18y 6xCritical points Find all critical points of the following functions. 22. f(x,y)=e8x2y224x28xy423E24E25E26E27E28EAnalyzing critical points Find the critical points of the following functions. Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum, a local minimum, or a saddle point. If the Second Derivative Test is inconclusive, determine the behavior of the function at the critical points. 29. f(x, y) = 4 + x4 + 3y430EAnalyzing critical points Find the critical points of the following functions. Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum, local minimum, or saddle point. Confirm your results using a graphing utility. 25.f(x,y)=x2+y24x+5Analyzing critical points Find the critical points of the following functions. Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum, local minimum, or saddle point. Confirm your results using a graphing utility. 26.f(x, y) = tan1 xy33E34EAnalyzing critical points Find the critical points of the following functions. Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum, local minimum, or saddle point. Confirm your results using a graphing utility. 29.f(x,y)=x1+x2+y2Analyzing critical points Find the critical points of the following functions. Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum, local minimum, or saddle point. Confirm your results using a graphing utility. 30.f(x,y)=x1x2+y237E38EAnalyzing critical points Find the critical points of the following functions. Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum, local minimum, or saddle point. Confirm your results using a graphing utility. 33.f(x, y) = yex ey40EInconclusive tests Show that the Second Derivative Test is inconclusive when applied to the following functions at (0, 0). Describe the behavior of the function at the critical point. 40.f(x, y) = x2y 3Inconclusive tests Show that the Second Derivative Test is inconclusive when applied to the following functions at (0, 0). Describe the behavior of the function at the critical point. 42.f(x, y) = sin(x2y2)Shipping regulations A shipping company handles rectangular boxes provided the sum of the height and the girth of the box does not exceed 96 in. (The girth is the perimeter of the smallest side of the box.) Find the dimensions of the box that meets this condition and has the largest volume.Cardboard boxes A lidless box is to be made using 2 m2 of cardboard. Find the dimensions of the box with the largest possible volume.Cardboard boxes A lidless cardboard box is to be made with a volume of 4 m3. Find the dimensions of the box that requires the least amount of cardboard.