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mathematics
calculus 4th

Questions and Answers of Calculus 4th

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
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.Does water density
Refer to Figure 26.Find the change in seawater density from A to B. 500 A Contour interval = 20 m 111 400 D 0 1 540 2 km
Refer to Figure 26.Estimate the average rate of change from A to B and from A to C. 500 A Contour interval = 20 m 111 400 D 0 1 540 2 km
Refer to Figure 26.Estimate the average rate of change from A to points i, ii, and iii. 500 A Contour interval = 20 m 111 400 D 0 1 540 2 km
Refer to Figure 26.Sketch the path of steepest ascent beginning at D. 500 A Contour interval = 20 m 111 400 D 0 1 540 2 km
Let temperature in 3-space be given by T(x, y, z) = x2/4 + y2/9 + z2. Draw isotherms corresponding to temperatures T = 0, 1, 4.
Let temperature in 3-space be given by T(x, y, z) = x2 − y2 − z. Draw isotherms corresponding to temperatures T = −1, 0, 1.
Let temperature in 3-space be given by T(x, y, z) = x2 − y2 − z2. Draw isotherms corresponding to temperatures T = −2, −1, 0, 1, 2.
The function ƒ(x, t) = t−1/2e−x2/t, whose graph is shown in Figure 27, models the temperature along a metal bar after an intense burst of heat is applied at its center point.(a) Sketch the
Let Write ƒ as a function ƒ(r, θ) in polar coordinates, and use this to find the level curves of ƒ. f(x, y) = X x + y for(x,y) = (0,0)
Prove that the functionis continuous at (0, 0). f(x, y) = < (2x - 1)(sin y) In 2 if xy # 0 if xy = 0
Prove that if ƒ(x) is continuous at x = a and g(y) is continuous at y = b, then F(x, y) = ƒ(x)g(y) is continuous at (a, b).
Consider the function (a) Show that as (x, y) → (0, 0) along any line y = mx, the limit equals 0.(b) Show that as (x, y) → (0, 0) along the curve y = x3, the limit does not equal 0, and
Compute the given partial derivatives. f(x, y) = sin(x - y), fy(0,7)
Use the contour map of ƒ(x, y) in Figure 8 to explain the following statements:(a) ƒy is larger at P than at Q, and ƒx is more negative at P than at Q.(b) ƒx(x, y) is decreasing as a function of
Estimate the partial derivatives at P of the function whose contour map is shown in Figure 9. 4 2 0 2 4 6 8 21 18- 15 12 -9 6 3 -X
Over most of the earth, a magnetic compass does not point to true (geographic) north; instead, it points at some angle east or west of true north. The angle D between magnetic north and true north is
Refer to Table 1.(a) Using difference quotients, approximate ∂ρ/∂T and ∂ρ/∂S at the points (S, T) = (30, 2), (32, 6), and (35, 10).(b) For fixed salinity S = 32, determine whether the
Compute the derivatives indicated f(x,y)= 3xy - 6xy4, 8 f 2f and x dy
Compute the derivatives indicated g(x,y) = - Hg
Compute the derivatives indicated h(u, v) = = u + 4v hvv (u, v)
Compute the derivatives indicated h(x, y) = ln(x +y), hxy (x, y)
Compute the derivatives indicated f(x, y) = xln(y), fyy(2,3)
Compute the derivatives indicated g(x,y) = xe-*", = xexy, gxy(-3,2)
Compute ƒxyxzy forUse a well-chosen order of differentiation on each term. f(x, y, z) = y sin(xz) sin(x + z) + (x + ) tany + x tan z+z \y-y-
LetWhat is the fastest way to show that ƒuvxyvu(x, y, u, v) = 0 for all (x, y, u, v)? f(x, y, u, v) = x + ev 3y + In(2 + u)
Compute the derivative indicated. annf (+) = (^'n)f
Compute the derivative indicated. g(x, y, z) = xyz6, 8xxyz
Compute the derivative indicated. F(r, s, t) = r(s + t), Frst
Compute the derivative indicated. xxn *(+/zx)-z/t_1 = (1x)n
Compute the derivative indicated. F(0,u,v) = sinh(uv +r), Fuue
Compute the derivative indicated. R(u, v, w) = v + w Ruvw
Compute the derivative indicated. g(x, y,z) = x + y +z, 8xyz
Compute ƒx and ƒy.ƒ(x, y) = 4xy3
Compute ƒx and ƒy.ƒ(x, y) = sin(xy)e−x−y
Compute ƒx and ƒy.ƒ(x, y) = ln(x2 + xy2)
Calculate ƒxxyz for f (x, y, z) = y sin(x + z).
Fix c > 0. Show that for any constants α, β, the function u(t, x) = sin(αct + β) sin(αx) satisfies the wave equation Ju at2 20 2
Find an equation of the tangent plane to the graph of ƒ(x, y) = xy2 − xy + 3x3y at P = (1, 3).
Suppose that ƒ(4, 4) = 3 and ƒx(4, 4) = fy(4, 4) = −1. Use the Linear Approximation to estimate ƒ(4.1, 4) and ƒ(3.88, 4.03).
Use a Linear Approximation of ƒ(x, y, z) = √x2 + y2 + z to estimate √7.12 + 4.92 + 69.5. Compare with a calculator value.
The plane z = 2x − y − 1 is tangent to the graph of z = ƒ(x, y) at P = (5, 3).(a) Determine ƒ(5, 3), ƒx(5, 3), and ƒy(5, 3).(b) Approximate ƒ(5.2, 2.9).
Figure 4 shows the contour map of a function ƒ(x, y) together with a path r(t) in the counterclockwise direction. The points r(1), r(2), and r(3) are indicated on the path. Let g(t) = ƒ (r(t)).
Jason earns S (h, c) = 20h (1 + c/100)1.5 dollars per month at a used car lot, where h is the number of hours worked and c is the number of cars sold. He has already worked 160 hours and sold 69
Compute d/dt ƒ(r(t)) at the given value of t. f(x,y) = x + e, r(t) = (3t-1, 1) at at t = 2
Compute d/dt ƒ(r(t)) at the given value of t. f(x, y, z) = xz - y, r(t) = (t,t, 1-t) att = -2
Compute d/dt ƒ(r(t)) at the given value of t. f(x,y) = xey - yex, r(t) = (e, Int) at t = 1
Compute d/dt ƒ(r(t)) at the given value of t. f(x, y) = tan y, X' r(t)= (cost, sint), t =
Compute the directional derivative at P in the direction of v. f(x, y) = xy4, P = (3,-1), v= 2i + j
Compute the directional derivative at P in the direction of v. f(x, y, z)=zx-xy, P= (1, 1, 1), v = (2,-1,2)
Compute the directional derivative at P in the direction of v. _f(x,y) = ex+,P= 5 2 VV2 2 2 v = 3, -4
Compute the directional derivative at P in the direction of v. f(x, y, z)= sin(xy+z), P = (0, 0, 0), v=j+ k
Find the unit vector e at P = (0, 0, 1) pointing in the direction along which ƒ(x, y, z) = xz + e−x2+y increases most rapidly.
Find an equation of the tangent plane at P = (0, 3, −1) to the surface with equation zet + e+ = xy +y-3
Let n ≠ 0 be an integer and r an arbitrary constant. Show that the tangent plane to the surface xn + yn + zn = r at P = (a, b, c) has equation a-x + by+c- = r
Let ƒ(x, y) = (x − y)ex. Use the Chain Rule to calculate ∂ƒ/∂u and ∂ƒ/∂v (in terms of u and v), where x = u − v and y = u + v. Since x First we calculate the Primary Derivatives: af f
Let ƒ(x, y, z) = x2y + y2z. Use the Chain Rule to calculate ∂ƒ/∂s and ∂ƒ/∂t (in terms of s and t), wherex = s + t, y = st, z = 2s − t
Let P have spherical coordinates Recall that x = ρ cos θ sin ϕ, y = ρ sin θ sin ϕ, z = ρ cos φ. af (p, 0, 4) = (2, 4, 4). Calculate 20 assuming that
Let g(u, v) = ƒ(u3 − v3, v3 − u3). Prove that g + U dv = 0
Let ƒ(x, y) = g(u), where u = x2 + y2 and g(u) is differentiable. Prove that 2 (1) + ( ) = 4u 2 N (dg) du
Calculate ∂z/∂x, where xez + zey = x + y.
Let ƒ(x, y) = x4 − 2x2 + y2 − 6y.(a) Find the critical points of ƒ and use the Second Derivative Test to determine whether they are a local minima or a local maxima.(b) Find the minimum value
Find the critical points of the function and analyze them using the Second Derivative Test. f(x, y) = x - 4xy + 2y
Find the critical points of the function and analyze them using the Second Derivative Test. f(x,y) = x + 2y xy
Find the critical points of the function. Then use the Second Derivative Test to determine whether they are local minima, local maxima, or saddle points (or state that the test fails). f(x, y) =
Find the critical points of the function. Then use the Second Derivative Test to determine whether they are local minima, local maxima, or saddle points (or state that the test fails). f(x, y) = ln x
Find the critical points of the function. Then use the Second Derivative Test to determine whether they are local minima, local maxima, or saddle points (or state that the test fails). f(x, y) = (x +
Find the critical points of the function. Then use the Second Derivative Test to determine whether they are local minima, local maxima, or saddle points (or state that the test fails). f(x, y) = x -
Find the critical points of the function. Then use the Second Derivative Test to determine whether they are local minima, local maxima, or saddle points (or state that the test fails). f(x,y) = (x -
Find the critical points of the function. Then use the Second Derivative Test to determine whether they are local minima, local maxima, or saddle points (or state that the test fails). -2 f(x, y) =
Show that ƒ(x, y) = x2 has infinitely many critical points (as a function of two variables) and that the Second Derivative Test fails for all of them. What is the minimum value of ƒ? Does ƒ(x, y)
Prove that the function ƒ(x, y) = 1/3 x3 + 2/3 y3/2 − xy satisfies ƒ(x, y) ≥ 0 for x ≥ 0 and y ≥ 0. (a) First, verify that the set of critical points of f is the parabola y = x and that
Let ƒ(x, y) = (x2 + y2)e−x2−y2. (a) Where does f take on its minimum value? Do not use calculus to answer this question. (b) Verify that the set of critical points of f consists of the origin
Use a computer algebra system to find a numerical approximation to the critical point ofApply the Second Derivative Test to confirm that it corresponds to a local minimum as in Figure 21. f(x, y) =
Which of the following domains are closed and which are bounded? (a) {(x, y) R: x + y 1} (b) {(x, y) = R: x + y 0, y > 0} (e) {(x, y) = R: 1 x 4,5 y 10} (f) {(x, y) = R : x > 0, x + y 10}
Determine the global extreme values of the function on the given set without using calculus. f(x, y) = x+y, 0x1, 0 y 1
Determine the global extreme values of the function on the given set without using calculus. f(x, y) = 2x-y, 0x 1, 0 y 3
Determine the global extreme values of the function on the given set without using calculus. f(x, y) = (x + y +1)-, 0x3, 0y5
Determine the global extreme values of the function on the given set without using calculus. f(x, y) = e-x-y x + y 1
Find the global minimum and maximum values of ƒ(x, y) on the specified polygon, and indicate where on the polygon they occur. (-3,0) (0,2) (1,0) (0,4) (A) (4, 11) (2,7) (4, 6) (3, 1) (B) (8, 7) (7,4)
Find the global minimum and maximum values of ƒ(x, y) on the specified polygon, and indicate where on the polygon they occur. (-3,0) (0,2) (1,0) (0,4) (A) (4, 11) (2,7) (4, 6) (3, 1) (B) (8,7) (7,4)
Find the global minimum and maximum values of ƒ(x, y) on the specified polygon, and indicate where on the polygon they occur. (-3,0) (0,2) (1,0) (0,4) (A) (4, 11) (2,7) (4, 6) (3, 1) (B) (8,7) (7,4)
Find the global minimum and maximum values of ƒ(x, y) on the specified polygon, and indicate where on the polygon they occur. (-3,0) (0,2) (1,0) (0,4) (A) (4, 11) (2,7) (4, 6) (3, 1) (B) (8,7) (7,4)
Show that ƒ(x, y) = xy does not have a global minimum or a global maximum on the domainExplain why this does not contradict Theorem 3. D= {(x,y): 0
Find a continuous function that does not have a global maximum on the domain D = {(x, y) : x + y ≥ 0, x + y ≤ 1}. Explain why this does not contradict Theorem 3.
Find the maximum of f(x,y) = x + y - x - y - xy
Find the maximum of ƒ(x, y) = y2 + xy − x2 on the square domain 0 ≤ x ≤ 2, 0 ≤ y ≤ 2.
Determine the global extreme values of the function on the given domain. f(x,y) = x - 2y, 0x1, 0y1
Determine the global extreme values of the function on the given domain. f(x, y) = 5x- 3y, y x-2, yz-x-2, y 3
Determine the global extreme values of the function on the given domain. f(x, y) = x + 2y, 0x1, 0 y 1
Determine the global extreme values of the function on the given domain. f(x, y)= x + xy + 2y, x, y 0, x+yl
Determine the global extreme values of the function on the given domain. f(x,y) = x + xy + y, x,y 0, x+y 1
Determine the global extreme values of the function on the given domain. f(x, y) = x + y - 3xy, 0x1, 0 y 1
Determine the global extreme values of the function on the given domain. f(x, y) = x + y - 2x - 4y, x>0, x 0, 0 y 3, y x
Determine the global extreme values of the function on the given domain. f (x, y) =(4y-x)e-x-y, x + y 2
The surface area of a right-circular cone of radius r and height h is S = πr √r2 + h2, and its volume is V = 1/3 πr2h. (a) Determine the ratio h/r for the cone with given surface area S and
In Example 1, we found the maximum of ƒ(x, y) = 2x + 5y on the ellipse (x/4)2 + (y/3)2 = 1. Solve this problem again without using Lagrange multipliers. First, show that the ellipse is parametrized
Find the point on the ellipse x + 6y + 3xy = 40
Use Lagrange multipliers to find the maximum area of a rectangle inscribed in the ellipse (Figure 15): (-x,y) (x, y) 2 a2 b2 + J2 = 1 (x, y) (x, y) X

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