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mathematics
calculus 4th

Calculus 4th Edition Jon Rogawski, Colin Adams, Robert Franzosa - Solutions

If S8,4 is a Riemann sum for a double integral over a rectangle R = [1, 5] × [2, 10] using a regular partition, what is the area of each subrectangle? How many subrectangles are there?
Estimate the double integral of a continuous function ƒ over the small rectangle R = [0.9, 1.1] × [1.9, 2.1] if ƒ(1, 2) = 4.
What is the integral of the constant function f(x, y) = 5 over the rectangle [-2,3] [2,4]?
What is the interpretation of takes on both positive and negative values on R? JJR f(x,y) dA if f(x,y)
Which of (a) or (b) is equal to 5 SS. f(x,y) dy dx?
For which of the following functions is the double integral over the rectangle in Figure 15 equal to zero? Explain your reasoning. (a) f(x, y) = xy (c) f(x, y) = sin x (b) f(x, y) = xy (d) f(x, y) = ex
Given n data points (x1, y1), . . . , (xn, yn), the linear least-squares fit is the linear function that minimizes the sum of the squares (Figure 26): E(m, b) = ; - f(x;))2 j=1 f(x) = mx + b Show that the minimum value of E occurs for m and b satisfying the two equations m x; + bn = V; j=1 m
Find the minimum of f (x, y, z) = x2 + y2 + z2 subject to the two constraints x + y + z = 1 and x + 2y + 3z = 6.
Suppose that z is defined implicitly as a function of x and y by the equation F(x, y, z) = xz2 + y2z + xy − 1 = 0. (a) Calculate Fx, Fy, Fz. (b) Use Eq. (7) to calculate z x and
Prove the Product Rule for Gradients in Theorem 1. THEOREM 1 Properties of the Gradient If f(x, y, z) and g(x, y, z) are differentiable and c is a constant, then (i) V(f + g)= Vf + Vg (ii) V(cf) = cVf (iii) Product Rule for Gradients: V(fg) = fVg+gVf (iv) Chain Rule for Gradients: If F(t) is a
Let u be a unit vector. Show that the directional derivative Du ƒ is equal to the component of ∇ƒ along u.
Let ƒ(x, y) = (xy)1/3.(a) Use the limit definition to show that ƒx(0, 0) = ƒy(0, 0) = 0.(b) Use the limit definition to show that the directional derivative Du ƒ(0, 0) does not exist for any unit vector u other than i and j.(c) Is f differentiable at (0, 0)?
Use the definition of differentiability to show that if ƒ(x, y) is differentiable at (0, 0) andthen f(0,0) = fx(0, 0) = fy(0, 0) = 0
This exercise shows that there exists a function that is not differentiable at (0, 0) even though all directional derivatives at (0, 0) exist. Define ƒ(x, y) = x2y/(x2 + y2) for (x, y) ≠ 0 and ƒ(0, 0) = 0.(a) Use the limit definition to show that Dv ƒ(0, 0) exists for all vectors v. Show that
Prove that if ƒ(x, y) is differentiable and ∇ƒ(x,y) = 0 for all (x, y), then ƒ is constant.
Prove the following Quotient Rule, where ƒ, g are differentiable: V (1) = gVf- fVg
A path r(t) = (x(t), y(t)) follows the gradient of a function ƒ(x, y) if the tangent vector r'(t) points in the direction of ∇ƒ for all t. In other words, r'(t) = k(t)∇ƒr(t) for some positive function k(t). In this case, r(t) crosses each level curve of ƒ(x, y) at a right angle.Show that if
Find a path of the form r(t) = (t, g(t)) passing through (1, 2) that follows the gradient of ƒ(x, y) = 2x2 + 8y2 (Figure 16). Hint: Use Separation of Variables. y 2 1+ K 1 X
Find the curve y = g(x) passing through (0, 1) that crosses each level curve of ƒ(x, y) = y sin x at a right angle. Using a computer algebra system, graph y = g(x) together with the level curves of ƒ.
ρ(S, T) is seawater density (kilograms per cubic meter) as a function of salinity S (parts per thousand) and temperature T (degrees Celsius). Refer to the contour map in Figure 25.Calculate the average rate of change of ρ with respect to S from B to C. Temperature T (C) 25+ 20- 15- 10- 5 1.02307
ρ(S, T) is seawater density (kilograms per cubic meter) as a function of salinity S (parts per thousand) and temperature T (degrees Celsius). Refer to the contour map in Figure 25.At a fixed level of salinity, is seawater density an increasing or a decreasing function of temperature? Temperature T
Let temperature in 3-space be given by T(x, y, z) = x2 + y2 − z. Draw isotherms corresponding to temperatures T = −2, −1, 0, 1, 2.
Compute the derivative indicated. u(x, t) = sech(x-t), Uxxx
Prove that there does not exist any function ƒ(x, y) such that Consider Clairaut’s Theorem. af x = xy and af dy || x.
Assume that ƒxy and ƒyx are continuous and that ƒyxx exists. Show that ƒxyx also exists and that ƒyxx = ƒxyx.
Show that u(x, t) = sin(nx) e−n2t satisfies the heat equation for any constant n: t || Ju -2
Find all values of A and B such that ƒ(x, t) = eAx+Bt satisfies Eq. (3). t || Ju 22
The functiondescribes the temperature profile along a metal rod at time t > 0 when a burst of heat is applied at the origin (see Example 11). A small bug sitting on the rod at distance x from the origin feels the temperature rise and fall as heat diffuses through the bar. Show that the bug feels
The Laplace operator Δ is defined by Δƒ = ƒxx + ƒyy. A function u(x, y) satisfying the Laplace equation Δu = 0 is called harmonic. Show that the following functions are harmonic: (a) u(x, y) = X (c) u(x, y) = tan- Y X (b) U(x, y) = e cosy (d) u(x, y) = In(x + y)
The Laplace operator Δ is defined by Δƒ = ƒxx + ƒyy. A function u(x, y) satisfying the Laplace equation Δu = 0 is called harmonic.Find all harmonic polynomials u(x, y) of degree 3, that is, u(x, y) = ax + bxy + cxy + dy.
The Laplace operator Δ is defined by Δƒ = ƒxx + ƒyy. A function u(x, y) satisfying the Laplace equation Δu = 0 is called harmonic.Show that if u(x, y) is harmonic, then the partial derivatives ∂u/∂x and ∂u/∂y are harmonic.
The Laplace operator Δ is defined by Δƒ = ƒxx + ƒyy. A function u(x, y) satisfying the Laplace equation Δu = 0 is called harmonic.Find all constants a, b such that u(x, y) = cos(ax)eby is harmonic.
Show that u(x, t) = sech2(x − t) satisfies the Korteweg–deVries equation (which arises in the study of water waves): 4u+ Uxxx +12uux = 0
This exercise shows that the hypotheses of Clairaut’s Theorem are needed. Let for (x, y) = (0,0) and f(0, 0) = 0. (a) Verify for (x, y) = (0,0): f(x, y) = xy- fx(x, y) = fy(x, y) = x - y x + y y(x4 + 4xy-y4) (x + y) x(x - 4xy - y) (x + y) (b) Use the limit definition of the partial derivative to
Use Eq. (6) to find the coefficients aT and aN as a function of t (or at the specified value of t).r(t) = (t, cos t, sin t) ar = a. T a-v |v|| an = a N = |la| - Jar|
Use Eq. (6) to find the coefficients aT and aN as a function of t (or at the specified value of t). ar = a. T a-v |v|| a = a N=|la| - Jar|
Use Eq. (14) to find the curvature of the curve given in polar form.ƒ(θ) = 2 cos θ K(0) = \(0) +2f'(0) = f(0)" (0)| ((0) + f'(0))/2 $K(0) = \(0)+2 f'(0) = f(0)" (0)| ((0) + f'(0))/2$
Use Eq. (5) to compute the curvature at the given point. k(t): [x' (t)y"(1) - y' (t)x" (1)| (x'(1) + y(1)2) /2
Use Eq. (5) to compute the curvature at the given point.(t cos t, sin t), t = π k(t): [x'(1)y"(t)- y'(t)x" (1)| (x'(1) + y(1)2) /2
The curve r(t) = (t − tanh t, sech t) is called a tractrix (see Exercise 100 in Section 11.1). = 5 ||r' (u)|| du is equal to s(t) = ln(cosht). (a) Show that s(t) = (b) Show that t = g(s) = ln(es + es 1) is an inverse of s(t) and verify that - r(s) = (tanh- ( e-2s) - - e-5, e-) - is an arc
Find the critical points of the function and analyze them using the Second Derivative Test. f(x,y) = ex+y - xey
Find the critical points of the function and analyze them using the Second Derivative Test.ƒ(x, y) = sin(x + y) − 1/2 (x + y2)
Prove that ƒ(x, y) = (x + 2y)exy has no critical points.
Find the global extrema of ƒ(x, y) = x3 − xy − y2 + y on the square [0, 1] × [0, 1].
Find the global extrema of ƒ(x, y) = 2xy − x − y on the domain {y ≤ 4, y ≥ x2}.
Find the maximum of ƒ(x, y, z) = xyz subject to the constraint g(x, y, z) = 2x + y + 4z = 1.
Use Lagrange multipliers to find the minimum and maximum values of ƒ(x, y) = 3x − 2y on the circle x2 + y2 = 4.
Find the minimum value of ƒ(x, y) = xy subject to the constraint 5x − y = 4 in two ways: using Lagrange multipliers and setting y = 5x − 4 in ƒ(x, y).
Find the minimum and maximum values of ƒ(x, y) = x2y on the ellipse 4x2 + 9y2 = 36.
Find the point in the first quadrant on the curve y = x + x−1 closest to the origin.
Find the extreme values of ƒ(x, y, z) = x + 2y + 3z subject to the two constraints x + y + z = 1 and x2 + y2 + z2 = 1.
Find the minimum and maximum values of ƒ(x, y, z) = x − z on the intersection of the cylinders x2 + y2 = 1 and x2 + z2 = 1 (Figure 5). X
Use Lagrange multipliers to find the dimensions of a cylindrical can with a bottom but no top, of fixed volume V with minimum surface area.
Given n nonzero numbers σ1, . . . , σn, show that the minimum value of subject to x + ... f(x1,...,Xn) = xr+ + Xn = 1 is c, where c ; (~3)= j=1 tox +
Determine the global extreme values of the function on the given domain. f(x, y) = x + 2xy, x + y 1
Find the maximum volume of a box inscribed in the tetrahedron bounded by the coordinate planes and the plane 1 1 x+2y+12=1 3
Find the volume of the largest box of the type shown in Figure 24, with one corner at the origin and the opposite corner at a point P = (x, y, z) on the paraboloid z = 1- X x 4 9 N with x, y, z 0 P y
Find the point on the plane z = x + y + 1 closest to the point P = (1, 0, 0). Minimize the square of the distance.
Show that the sum of the squares of the distances from a point P = (c, d) to n fixed points (a1, b1), . . . ,(an, bn) is minimized when c is the average of the x-coordinates ai and d is the average of the y-coordinates bi.
Show that the rectangular box (including the top and bottom) with fixed volume V = 27 m3 and smallest possible surface area is a cube (Figure 25). X y Z
Consider a rectangular box B that has a bottom and sides but no top and has minimal surface area among all boxes with fixed volume V = 64 m3.(a) Do you think B is a cube as in the solution to Exercise 54? If not, how would its shape differ from a cube?Data From Exercise 54Show that the rectangular
Find three positive numbers that sum to 150 with the greatest possible product of the three.
A 120-m long fence is to be cut into pieces to make three enclosures, each of which is square. How should the fence be cut up in order to minimize the total area enclosed by the fence?
A box with a volume of 8 m3 is to be constructed with a gold-plated top, silver-plated bottom, and copper-plated sides. If gold plate costs $120 per square meter, silver plate costs $40 per square meter, and copper plate costs $10 per square meter, find the dimensions that will minimize the cost of
Find the maximum volume of a cylindrical can such that the sum of its height and its circumference is 120 cm.
The power (in microwatts) of a laser is measured as a function of current (in milliamps). Find the linear least-squares fit (Exercise 60) for the data points.Data From Exercise 60Given n data points (x1, y1), . . . , (xn, yn), the linear least-squares fit is the linear function Current (milliamps)
Let A = (a, b) be a fixed point in the plane, and let ƒA(P) be the distance from A to the point P = (x, y). For P ≠ A, let eAP be the unit vector pointing from A to P (Figure 27): y Distance f(x, y) A = (a, b) e AP P = (x, y) - X
In this exercise, we prove that for all x, y ≥ 0: a + b 1 1.40 y B where a 1 and 1 are numbers such that a +B- = 1. To do this, we prove that the function f(x,y) = ax + _xy xy satisfies f(x, y) 0 for all x, y 0. (a) Show that the set of critical points of f(x, y) is the curve y = x-1 (Figure
The following problem was posed by Pierre de Fermat: Given three points A = (a1, a2), B = (b1, b2), and C = (c1, c2) in the plane, find the point P = (x, y) that minimizes the sum of the distances f(x, y) = AP + BP + CP Let e, f, g be the unit vectors pointing from P to the points A, B, C as in
Find the extreme values of ƒ(x, y) = x2 + 2y2 subject to the constraint g(x, y) = 4x − 6y = 25. (a) Show that the Lagrange equations yield 2x = 42, 4y = -62. (b) Show that if x = 0 or y = 0, then the Lagrange equations give x = y = 0. Since (0,0) does not satisfy the constraint, you may assume
Consider the problem of minimizing ƒ(x, y) = x subject to g(x, y) = (x − 1)3 − y2 = 0.(a) Show, without using calculus, that the minimum occurs at P = (1, 0).(b) Show that the Lagrange condition ∇ƒP = λ∇gP is not satisfied for any value of λ.(c) Does this contradict Theorem 1? THEOREM
Goods 1 and 2 are available at dollar prices of p1 per unit of Good 1 and p2 per unit of Good 2. A utility function U(x1, x2) is a function representing the utility or benefit of consuming xj units of good j. The marginal utility of the jth good is ∂U/∂xj, the rate of increase in utility per
Consider the utility function U(x1, x2) = x1x2 with budget constraint p1x1 + p2x2 = c.(a) Show that the maximum of U(x1, x2) subject to the budget constraint is equal to c2/(4p1 p2).(b) Calculate the value of the Lagrange multiplier λ occurring in (a).(c) Prove the following interpretation: λ is
This exercise shows that the multiplier λ may be interpreted as a rate of change in general. Assume that the maximum of ƒ(x, y) subject to g(x, y) = c occurs at a point P. Then P depends on the value of c, so we may write P = (x(c), y(c)) and we have g(x(c), y(c)) = c. (a) Show that Vg(x(c),
Let B > 0. Show that the maximum of subject to the constraints x + this to conclude that f(x,...,xn) = X1 X2 Xn for all positive numbers a,..., an. . + xn = B and xj 0 for j = 1,. n occurs for x = ... ... (aa... an) /n ... a +. + An n =Xn= B/n. Use
Let B > 0. Show that the maximum of ƒ(x1, . . . , xn) = x1 + · · · + xn subject to x21 + · · · + x2n = B2 is √nB. Conclude that for all numbers a,..., an lal++lanl n(a + + a) / 1/2
Let r = √x21 + · · · + x2n and let g(r) be a function of r. Prove the formulas dg ; Xi -gr, r 02g ? || x? r28rr + p - x p.3 -gr
Prove that if g(r) is a function of r as in Exercise 49, thenData From Exercise 49Let r = √x21 + · · · + x2n and let g(r) be a function of r. Prove the formulas ag ax + + 8g = grr + x 'n n - 1 -gr r
The Laplace operator is defined by Δƒ = ƒxx + ƒyy. A function ƒ(x, y) satisfying the Laplace equation Δƒ = 0 is called harmonic. A function ƒ(x, y) is called radial if ƒ(x, y) = g(r), where r = √x2 + y2.Use Eq. (12) to prove that in polar coordinates (r, θ), 2x + fx a + fy a y ar
Use Eq. (13) to show that ƒ(x, y) = ln r is harmonic. 1 Af = frr + fee + = fr 1 2
Verify that ƒ(x, y) = x and ƒ(x, y) = y are harmonic using both the rectangular and polar expressions for Δƒ.
Verify that ƒ(x, y) = tan−1 y/x is harmonic using both the rectangular and polar expressions for Δƒ.
Use the Product Rule to show thata af fm + fr = r 0 (1) r rUse this formula to show that if f is a radial harmonic function, then rƒr = C for some constant C. Conclude that ƒ(x, y) = C ln r + b for some constant b. a af fm + fr = r 0 (1) r r
Use a computer algebra system to produce a contour plot of ƒ(x, y) = x2 − 3xy + y − y2 together with its gradient vector field on the domain [−4, 4] × [−4, 4].
Find a function ƒ(x, y, z) such that ∇ƒ is the constant vector (1, 3, 1).
Find a function f (x, y, z) such that ∇ƒ = (2x, 1, 2).
Find a function ƒ(x, y, z) such that ∇ƒ= (x, y2, z3).
Find a function (x, y, z) such that ∇ƒ= (z, 2y, x).
Find a function ƒ(x, y) such that ∇ƒ = (y, x).
Show that there does not exist a function ƒ(x, y) such that ∇ƒ = (y2, x). Use Clairaut’s Theorem ƒxy = ƒyx. Clairaut's Theorem fxyy = fyxy = fyyx
Let Δƒ = ƒ(a + h, b + k) − ƒ(a, b) be the change in ƒ at P = (a, b). Set Δv =(h, k). Show that the Linear Approximation can be written f ~ Vfp P
Use Eq. (6) to estimate Af = f(3.53, 8.98) - f(3.5,9)
Find a unit vector n that is normal to the surface z2 − 2x4 − y4 = 16 at P = (2, 2, 8) that points in the direction of the xy-plane (in other words, if you travel in the direction of n, you will eventually cross the xy-plane).
Suppose, in the previous exercise, that a particle located at the point P = (2, 2, 8) travels toward the xy-plane in the direction normal to the surface.(a) Through which point Q on the xy-plane will the particle pass?(b) Suppose the axes are calibrated in centimeters. Determine the path r(t) of
Let ƒ(x, y) = tan−1 x/y and u = (√2/2 , √2/2). (a) Calculate the gradient of f. (b) Calculate Duf(1, 1) and Du(3, 1). (c) Show that the lines y = mx for m # 0 are level curves for f. (d) Verify that Vfp is orthogonal to the level curve through P for P = (x, y) # (0,0).
Suppose that the intersection of two surfaces F(x, y, z) = 0 and G(x, y, z) = 0 is a curve C, and let P be a point on C. Explain why the vector v = ∇FP × ∇GP is a direction vector for the tangent line to C at P.
Let C be the curve of intersection of the spheres x2 + y2 + z2 = 3 and (x − 2)2 + (y − 2)2 + z2 = 3. Use the result of Exercise 63 to find parametric equations of the tangent line to C at P = (1, 1, 1).Data From Exercise 63Suppose that the intersection of two surfaces F(x, y, z) = 0 and
Let C be the curve obtained as the intersection of the two surfaces x3 + 2xy + yz = 7 and 3x2 − yz = 1. Find the parametric equations of the tangent line to C at P = (1, 2, 1).
Prove the linearity relations for gradients: (a) V(f + g) = Vf + Vg (b) V(cf) = cVf
Prove the Chain Rule for Gradients in Theorem 1. THEOREM 1 Properties of the Gradient If f(x, y, z) and g(x, y, z) are differentiable and c is a constant, then (i) V(f+g) = Vf + Vg (ii) V(cf) = cVf (iii) Product Rule for Gradients: V(fg) = fVg + gVf (iv) Chain Rule for Gradients: If F(t) is a

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