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mathematics
calculus 4th

Questions and Answers of Calculus 4th

Calculate ∂P/∂T and ∂P/∂V, where pressure P, volume V, and temperature T are related by the Ideal Gas Law, PV = nRT (R and n are constants).
Given ƒ(x, y) =√x2 − y2 /x + 3: (a) Sketch the domain of f. (b) Calculate f(3, 1) and f(-5,-3). (c) Find a point satisfying f(x, y) = 1.
Find the domain and range of: (a) f(x, y, z)= x-y + y-z (b) f(x, y) = In(4x - y)
Sketch the graph ƒ(x, y) = x2 − y + 1 and describe its vertical and horizontal traces.
Use a graphing utility to draw the graph of the function cos(x2 + y2)e1−xy in the domains [−1, 1] × [−1, 1], [−2, 2] × [−2, 2], and [−3, 3] × [−3, 3], and explain its behavior.
Match the functions (a)–(d) with their graphs in Figure 1. (a) f(x, y) = x + y (b) f(x,y) = x + 4y (c) f(x, y) = sin(4xy)e-x-y (d) f(x, y) = sin(4x)e-x-y
Referring to the contour map in Figure 2:(a) Estimate the average rate of change of elevation from A to B and from A to D.(b) Estimate the directional derivative at A in the direction of v.(c) What
Describe the level curves of: (a) f(x, y) = 4x-y (c) f(x,y) = 3x - 4y (b) f(x, y) = ln(4x - y) (d) f(x, y) = x + y
Match each function (a)–(c) with its contour graph (i)–(iii) in Figure 3: (a) f(x,y) = xy (b) f(x, y) = exy (c) f(x, y) = sin(xy)
Evaluate the limit or state that it does not exist. lim (xy + y) (x,y)(1,-3)
Evaluate the limit or state that it does not exist. lim (x,y) (1,-3) In(3x + y)
Evaluate the limit or state that it does not exist. xy + xy lim (x,y) (0,0) x + y
Evaluate the limit or state that it does not exist. xy + xy lim (x,y) (0,0) x+y+
Evaluate the limit or state that it does not exist. lim (x,y) (1,-3) (2x + y)e-x+y
Evaluate the limit or state that it does not exist. lim (x,y)(0,2) (ex - 1)(e - 1) X
LetUse polar coordinates to show that f (x, y) is continuous at all (x, y) if p > 2 but is discontinuous at (0, 0) if p ≤ 2. (xy)p f(x,y)=x4 +34 0 (x, y) = (0, 0) (x, y) = (0, 0)
Calculate ƒx(1, 3) and ƒy(1, 3) for (x, y) = √7x + y2.
Compute ƒx and ƒy. f(x, y) = 2x + y
Let P = (a, b) be a critical point of ƒ(x, y) = x2 + y4 − 4xy. (a) First use fx(x, y) = 0 to show that a = 2b. Then use fy(x,y) = 0 to show that P = (0,0), (2 2, 2), or (-2 2,-2). (b) Referring to
Find the critical points of the functions f(x, y) = x + 2y - 4y +6x, g(x, y)=x-12xy +y
Find the critical points ofUse the contour map in Figure 19 to determine their nature (local minimum, local maximum, or saddle point). f(x, y) = 8y4+x + xy - 3y-y
Use the contour map in Figure 20 to determine whether the critical points A, B,C, D are local minima, local maxima, or saddle points. y 2 0 -2. 1 A D 0 0 0 B 'C 23 2 x
Let ƒ(x, y) = y2x − yx2 + xy. (a) Show that the critical points (x, y) satisfy the equations y(y - 2x + 1) = 0, x(2y-x + 1) = 0 (b) Show that f has three critical points where x = 0 or y = 0 (or
Show that ƒ(x, y) = √x2 + y2 has one critical point P and that ƒ is nondifferentiable at P. Does f have a minimum, maximum, or saddle point at P?
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). f(x,y) = 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)= 4x -
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 + 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) = ex-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) =
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).ƒ(x, y) = ex
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) =
Suppose that the maximum of ƒ(x, y) subject to the constraint g(x, y) = 0 occurs at a point P = (a, b) such that ∇ƒP ≠ 0. Which of the following statements is true? (a) Vfp is tangent to g(x,
Figure 10 shows a constraint g(x, y) = 0 and the level curves of a function ƒ. In each case, determine whether ƒ has a local minimum, a local maximum, or neither at the labeled point. A Vf 4321
On the contour map in Figure 11:(a) Identify the points where ∇ƒ = λ∇g for some scalar λ.(b) Identify the minimum and maximum values of ƒ(x, y) subject to g(x, y) = 0. y -6-212 6 6 2-2-6
Use the method of Lagrange multipliers unless otherwise stated. Find the extreme values of the function f(x, y) = 2x + 4y subject to the constraint g(x,y) = x + y - 5 = 0. (a) Show that the Lagrange
Apply the method of Lagrange multipliers to the function ƒ(x, y) = (x2 + 1)y subject to the constraint x2 + y2 = 5. First show that y ≠ 0; then treat the cases x = 0 and x ≠ 0 separately.
Find the minimum and maximum values of the function subject to the given constraint. f(x, y) = 2x + 3y, x + y = 4
Find the minimum and maximum values of the function subject to the given constraint. f(x, y) = x + y, 2x + 3y = 6
Find the minimum and maximum values of the function subject to the given constraint. f(x,y) = 4x +9y, xy = 4
Find the minimum and maximum values of the function subject to the given constraint. f(x, y) =xy, 4x +9y = 32
Find the minimum and maximum values of the function subject to the given constraint. f(x,y) = xy + x + y, xy = 4
Find the minimum and maximum values of the function subject to the given constraint. f (x, y) = x + y, x + y4 = 1
Find the minimum and maximum values of the function subject to the given constraint. f(x, y) = xy4, x + 2y = 6
Find the minimum and maximum values of the function subject to the given constraint. f(x, y, z) = 3x + 2y + 4z, x + 2y + 6z = 1
Find the minimum and maximum values of the function subject to the given constraint. f(x,y, z)=x-y-z, x - y + z =0
Find the minimum and maximum values of the function subject to the given constraint. f(x, y,z)=xy + 2z, x + y + z = 36
Find the minimum and maximum values of the function subject to the given constraint. f(x, y, z) = x + y + z, x + 3y + 2z = 36
Find the minimum and maximum values of the function subject to the given constraint. f(x, y, z) = xy + xz, x + y + z = 4
Let (a) Show that there is a unique point P = (a, b) on g(x, y) = 1 where ∇ƒP = λ∇gP for some scalar λ.(b) Refer to Figure 13 to determine whether ƒ(P) is a local minimum or a local maximum
Find the point (a, b) on the graph of y = ex where the value ab is the least.
Find the rectangular box of maximum volume if the sum of the lengths of the edges is 300 cm.
Let ƒ(x, y, z) = x2y3 + z4 and x = s2, y = st2, and z = s2t. f f f (a) Calculate the primary derivatives z. (b) Calculate (c) Compute z. Os' Os' as af - using the Chain Rule: af af ax s f
Let ƒ(x, y) = x cos(y) and x = u2 + v2 and y = u − v. af af (a) Calculate the primary derivatives x' dy (b) Use the Chain Rule to calculate df/av. Leave the answer in terms of both the dependent
Use the Chain Rule to calculate the partial derivatives. Express the answer in terms of the independent variables. f. s r -; f(x, y, z) = xy + z2, x = s, y = 2rs, z = r2
Use the Chain Rule to calculate the partial derivatives. Express the answer in terms of the independent variables. af af r t *; f(x,y,z) = xy + 2, x = r + s - 2t, y = 3rt, z = s2
Use the Chain Rule to calculate the partial derivatives. Express the answer in terms of the independent variables. g dg -; g(0,) = tan(0 + ), 0 = xy, = x + y
Use the Chain Rule to calculate the partial derivatives. Express the answer in terms of the independent variables. dv' dw * R(x,y) = x - 2y), x = w2, y = = y
Use the Chain Rule to calculate the partial derivatives. Express the answer in terms of the independent variables. OF -; F(u, v) = eu+v, u = x, v = xy
Use the Chain Rule to calculate the partial derivatives. Express the answer in terms of the independent variables. of . ; (x, y) = x + y, x = eu+v, y = u + v
Use the Chain Rule to calculate the partial derivatives. Express the answer in terms of the independent variables. dh t -; h(x, y) = X x = tt, y = tt " y
Use the Chain Rule to calculate the partial derivatives. Express the answer in terms of the independent variables. af 20 ; f(x,y,z) = xy z?, x = r cos 0, y = cos0, z = r
Use the Chain Rule to evaluate the partial derivative at the point specified. /u and /v at (u, v) = (1,1), where f(x, y, z) = x + yz, x = u + v, y = u + v, z , Z = uv
Use the Chain Rule to evaluate the partial derivative at the point specified. af/as at (r, s) = (1,0), where f(x, y) = ln(xy), x = 3r + 2s, y = 5r+ 3s
Use the Chain Rule to evaluate the partial derivative at the point specified. g/80 at (r, 0) = (2 2, 4), where g(x, y) = 1/(x + y), x = r cos 0, y = r sin
Use the Chain Rule to evaluate the partial derivative at the point specified. dg/ds at s = 4, where g(x, y) = x - y, x = s + 1, y = 1 - 2s
Use the Chain Rule to evaluate the partial derivative at the point specified. - g/u at (u, v) = (0, 1), where g(x, y) = x y, x = e cos v, y = e" sin v
Use the Chain Rule to evaluate the partial derivative at the point specified. h at (q, r) = (3, 2), where h(u, v) = ue', u = q, v = qr
Given ƒ(x, y) and y = y(x), we can define a composite function g(x) = ƒ(x, y(x)). af (a) Show that g'(x) = + af x y'(x). (b) Let f(x, y) = x xy and y(x) = 1 x. With g(x) = f(x, y(x)), use the
Let ƒ(x, y) = 4 − x2y2 + e2x and y(x) = ex/x . Define g(x) = ƒ(x, y(x)).(a) Use the derivative formula from Exercise 17(a) to prove that g (x) = 0 and therefore that g is a constant function.Data
The functions ƒ(x, y) = x2 + y2 and g(x, y) = x2 − y2 both have a critical point at (0, 0). How is the behavior of the two functions at the critical point different?
Identify the points indicated in the contour maps as local minima, local maxima, saddle points, or neither (Figure 16). 3 3 0-1 -3 0 13 -3-10 -10 10 -6 -22 0 6
Let ƒ(x, y) be a continuous function on a domain D in R2. Determine which of the following statements are true: (a) If D is closed and bounded, then f takes on a maximum value on D. (b) If D is
Let ƒ(x, y) = xy2 and r(t) = (1/2t2, t3). (a) Calculate Vf and r' (t). d (b) Use the Chain Rule for Paths to evaluate f(r(t)) at t = 1 and t = - 1.
Let ƒ(x, y) = exy and r(t) = (t3, 1 + t).(a) Calculate ∇ƒ and r'(t).(b) Use the Chain Rule for Paths to calculate d/dt ƒ(r(t)).(c) Write out the composite ƒ(r(t)) as a function of t and
Figure 14 shows the level curves of a function ƒ(x, y) and a path r(t), traversed in the direction indicated. State whether the derivative d /dt ƒ(r(t)) is positive, negative, or zero at points
Let ƒ(x, y) = x2 + y2 and r(t) = (cos t, sin t).(a) Find d/dtƒ(r(t)) without making any calculations. Explain.(b) Verify your answer to (a) using the Chain Rule.
Calculate the gradient. f(x, y) = cos(x + y)
Calculate the gradient. g(x, y) = X x + y
Calculate the gradient.h(x, y, z) = xyz−3
Calculate the gradient.r(x, y, z,w) = xzeyw
Use the Chain Rule to calculate d/dt ƒ(r(t)) at the value of t given. f(x, y) = 3x - 7y, r(t) = (cost, sin t), t = 0
Use the Chain Rule to calculate d/dt ƒ(r(t)) at the value of t given.ƒ(x, y) = 2x + 3y, r(t) =(t3, t2), t = −2
Use the Chain Rule to calculate d/dt ƒ(r(t)) at the value of t given.ƒ(x, y) = x2 − 3xy, r(t) = (cos t, sin t), t = 0
Use the Chain Rule to calculate d/dt ƒ(r(t)) at the value of t given.ƒ(x, y) = x2 − 3xy, r(t) = (cos t, sin t), t = π/2
Use the Chain Rule to calculate d/dt ƒ(r(t)) at the value of t given. (x, y) = cos(y x), _r(t) = (e, e), - t = ln 3
Use the Chain Rule to calculate d/dt ƒ(r(t)) at the value of t given. f(x, y) = cos(y x), r(t) = (e,e), - t = ln 3
Use the Chain Rule to calculate d/dt ƒ(r(t)) at the value of t given. f(x, y) = x = xy, r(t) = (R R - 41), t = 4
Use the Chain Rule to calculate d/dt ƒ(r(t)) at the value of t given. f(x, y) = 3xe, r(t) = (21, 1 2t), t = 0
Use the Chain Rule to calculate d/dt ƒ(r(t)) at the value of t given. (x,y)=lnx+lny, _r(t) = (cost, t), t = 4/
Use the Chain Rule to calculate d/dt ƒ(r(t)) at the value of t given. g(x, y, z) = xye, _r(t) = (P, P, t 1), t = 1
Let ƒ(x, y) = xy, where x = uv and y = u + v.(a) What are the primary derivatives of ƒ?(b) What are the independent variables?
Suppose that ƒ(u, v) = uev, where u = rs and v = r + s.The composite function ƒ(u, v) is equal to:(a) Rser+s (b) Res (c) Rsers
Suppose that ƒ(u, v) = uev, where u = rs and v = r + s.What is the value of ƒ(u, v) at (r, s) = (1, 1)?
According to the Chain Rule, ∂ ƒ/∂r is equal to (choose the correct answer): (a) (b) () af ax r of ax f r af r + + + f dx af af as
Suppose that x, y, z are functions of the independent variables u, v,w. Which of the following terms appear in the Chain Rule expression for ∂ ƒ/∂w? (a) (b) (c)
With notation as in the previous exercise, does ∂x/∂v appear in the Chain Rule expression for ∂ƒ /∂u?

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