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mathematics
calculus 4th

Questions and Answers of Calculus 4th

Find the point (x0, y0) on the line 4x + 9y = 12 that is closest to the origin.
Show that the point (x0, y0) closest to the origin on the line ax + by = c has coordinates Xo = ac a + b2' yo bc a + b2
Find the maximum value of constants. f(x, y) = xayb for x 0, y 0 on the line x + y = 1, where a, b > 0 are
Show that the maximum value of ƒ(x, y) = x2y3 on the unit circle is 6/25 √3/5.
Find the maximum value of ƒ(x, y) = xayb for x ≥ 0, y ≥ 0 on the unit circle, where a, b > 0 are constants.
Find the maximum value of ƒ(x, y, z) = xaybzc for x, y, z ≥ 0 on the unit sphere, where a, b, c > 0 are constants.
Show that the minimum distance from the origin to a point on the plane ax + by + cz = d is |d| a + b +c
Antonio has $5.00 to spend on a lunch consisting of hamburgers ($1.50 each) and french fries ($1.00 per order). Antonio’s satisfaction from eating x1 hamburgers and x2 orders of french fries is
Let Q be the point on an ellipse closest to a given point P outside the ellipse. It was known to the Greek mathematician Apollonius (third century bce) that PQ is perpendicular to the tangent to the
In a contest, a runner starting at A must touch a point P along a river and then run to B in the shortest time possible (Figure 17). The runner should choose the point P that minimizes the total
Let V be the volume of a can of radius r and height h, and let S be its surface area (including the top and bottom).Find r and h that minimize S subject to the constraint V = 54π.
Show that for both of the following two problems, P = (r, h) is a Lagrange critical point if h = 2r:• Minimize surface area S for fixed volume V.• Maximize volume V for fixed surface area S .Then
Figure 19 depicts a tetrahedron whose faces lie in the coordinate planes and in the plane with equation The volume of the tetrahedron is given by V = 1/6 abc. Find the minimum value of V among all
With the same set-up as in the previous problem, find the plane that minimizes V if the plane is constrained to pass through a point P = (α, β, γ) with α, β, γ > 0.
Show that the Lagrange equations for ƒ(x, y) = x + y subject to the constraint g(x, y) = x + 2y = 0 have no solution. What can you conclude about the minimum and maximum values of f subject to g =
Show that the Lagrange equations for ƒ(x, y) = 2x + y subject to the constraint g(x, y) = x2 − y2 = 1 have a solution but that f has no min or max on the constraint curve. Does this contradict
Let L be the minimum length of a ladder that can reach over a fence of height h to a wall located a distance b behind the wall. (a) Use Lagrange multipliers to show that L = (h/3 + b2/3 3/2 (Figure
Find the maximum value of ƒ(x, y, z) = xy + xz + yz − xyz subject to the constraint x + y + z = 1, for x ≥ 0, y ≥ 0, z ≥ 0.
Find the maximum of ƒ(x, y, z) = z subject to the two constraints x2 + y2 = 1 and x + y + z = 1.
Find the point lying on the intersection of the plane x + 1/2 y + 1/4 z = 0 and the sphere x2 + y2 + z2 = 9 with the greatest z-coordinate.
Find the maximum of ƒ(x, y, z) = x + y + z subject to the two constraints x2 + y2 + z2 = 9 and 1/4 x2 + 1/4 y2 + 4z2 = 9.
The cylinder x2 + y2 = 1 intersects the plane x + z = 1 in an ellipse. Find the point on such an ellipse that is farthest from the origin.
Find the minimum and maximum of ƒ(x, y, z) = y + 2z subject to two constraints, 2x + z = 4 and x2 + y2 = 1.
Find the minimum value of ƒ(x, y, z) = x2 + y2 + z2 subject to two constraints, x + 2y + z = 3 and x − y = 4.
Suppose that both ƒ(x, y) and the constraint function g(x, y) are linear. Use contour maps to explain why ƒ(x, y) does not have a maximum subject to g(x, y) = 0 unless g = aƒ + b for some
A baseball player hits the ball and then runs down the first base line at 20 ft/s. The first baseman fields the ball and then runs toward first base along the second base line at 18 ft/s as in Figure
Jessica and Matthew are running toward the point P along the straight paths that make a fixed angle of θ (Figure 8). Suppose that Matthew runs with velocity va meters per second and Jessica with
Two spacecraft are following paths in space given by r1 = (sin t, t, t2) and r2 = (cos t, 1 − t, t3). If the temperature for points in space is given by T(x, y, z) = x2y(1 − z), use the Chain
The Law of Cosines states that c2 = a2 + b2 − 2ab cos θ, where a, b, c are the sides of a triangle and θ is the angle opposite the side of length c. (a) Compute 20/da, 30/ab, and 20/ac using
Let u = u(x, y), and let (r, θ) be polar coordinates. Verify the relationCompute the right-hand side by expressing uθ and ur in terms of ux and uy. 1 2 ||ull = up + p2
Let u(r, θ) = r2 cos2 θ. Use Eq. (8) to compute ΙΙ∇uΙΙ2. Then compute ΙΙ∇uΙΙ2 directly by observing that u(x, y) = x2, and compare. I + "= "All|
Let x = s + t and y = s − t. Show that for any differentiable function ƒ(x, y), 2 (2) -- = af af s t
Express the derivativeswhere (ρ, θ, φ) are spherical coordinates. af of af ' ' , in terms of af af af ' z.
Calculate ∂z/∂x and ∂z/∂y at the points (3, 2, 1) and (3, 2, −1), where z is defined implicitly by the equation z4 + z2x2 − y − 8 = 0.
Calculate the partial derivative using implicit differentiation. z dx' xy + yz+xz = 10
Calculate the partial derivative using implicit differentiation. dw z. xw + w + wz + 3yz = 0
Calculate the partial derivative using implicit differentiation. z. exy + sin(xz) + y = 0
Calculate the partial derivative using implicit differentiation. r at and t ar' r2 = tes/r
Calculate the partial derivative using implicit differentiation. w 1 w2 + x2 + 1 w2 +y2 = 1 at (x, y, w) = (1,1,1)
Calculate the partial derivative using implicit differentiation. au/aT and OT/JU, (TU-V) In(W- UV) = In 2 at (T, U, V, W) = (1, 1, 2, 4)
Let r = (x, y, z) and er =r/ ΙΙrΙΙ. Show that if a function ƒ(x, y, z) = F(r) depends only on the distance from the origin r = ΙΙrΙΙ= √x2 + y2 + z2, then Vf = F'(r)er
Let ƒ(x, y, z) = e−x2−y2−z2 = e−r2 , with r as in Exercise 35. Compute ∇ƒ directly and using Eq. (9).Data From Exercise 35Let r = (x, y, z) and er =r/ ΙΙrΙΙ. Show that if a function
Use Eq. (9) to compute
Use Eq. (9) to compute ∇(ln r). Vf = F'(r)er
Figure 9 shows the graph of the equation Vf = F'(r)er
For all x > 0, there is a unique value y = r(x) that solves the equation y3 + 4xy = 16. (a) Show that dy/dx = 4y/(3y + 4x). (b) Let g(x) = f(x, r(x)), where f(x, y) is a function satisfying
The pressure P, volume V, and temperature T of a van der Waals gas with n molecules (n constant) are related by the equationwhere a, b, and R are constant. Calculate ∂P/∂T and ∂V/∂P. (P + 1 )
When x, y, and z are related by an equation F(x, y, z) = 0, we sometimes write (∂z/∂x)y in place of ∂z/∂x to indicate that in the differentiation, z is treated as a function of x with y held
Show that if ƒ(x) is differentiable and c ≠ 0 is a constant, then u(x, t) = ƒ(x − ct) satisfies the so-called advection equation +c- t = 0
A function ƒ(x, y, z) is called homogeneous of degree n if ƒ(λx, λy, λz) = λn ƒ(x, y, z) for all λ ∈ R.Show that the following functions are homogeneous and determine their degree: (a) f(x,
Prove that if ƒ(x, y, z) is homogeneous of degree n, then ƒx(x, y, z) is homogeneous of degree n − 1. Either use the limit definition or apply the Chain Rule to ƒ(λx, λy, λz).
Prove that if ƒ(x, y, z) is homogeneous of degree n, thenLet F(t) = ƒ(tx, ty, tz) and calculate F'(1) using the Chain Rule. af + x af af z +2 =nf
Verify Eq. (11) for the functions in Exercise 44.Data From Exercise 44A function ƒ(x, y, z) is called homogeneous of degree n if ƒ(λx, λy, λz) = λn ƒ(x, y, z) for all λ ∈ R.Show that the
Suppose that ƒ is a function of x and y, where x = g(t, s), y = h(t, s). Show that ƒtt is equal to fxx x at +2 () () at + fyy at + fx 22x + fy 12 12
Use the Chain Rule to calculate d/dt ƒ(r(t)) at the value of t given. g(x, y, z) = xyz, r(t) = (e, t, 1), t = 1
Use the Chain Rule to calculate d/dt ƒ(r(t)) at the value of t given. g(x, y, z, w) = x + 2y + 3z +5w, r(t) = (1,t,t,t 2), t = 1
Calculate the directional derivative in the direction of v at the given point. Remember to use a unit vector in your directional derivative computation. f(x,y) = x + y, v = (4,3), P = (1, 2)
Calculate the directional derivative in the direction of v at the given point. Remember to use a unit vector in your directional derivative computation. f(x, y) = xy x, v=i-j, P=(2,-1) -
Calculate the directional derivative in the direction of v at the given point. Remember to use a unit vector in your directional derivative computation. f(x,y)=xy, v=i+j, P= (1,3)
Calculate the directional derivative in the direction of v at the given point. Remember to use a unit vector in your directional derivative computation. f(x, y) = sin(x - y), v= (1, 1), P = ( 73, 7)
Calculate the directional derivative in the direction of v at the given point. Remember to use a unit vector in your directional derivative computation. f(x, y) = tan-(xy), v= (1, 1), P = (3, 4)
Calculate the directional derivative in the direction of v at the given point. Remember to use a unit vector in your directional derivative computation. f(x, y) = exy-y V = = (12,-5), P = (2,2)
Calculate the directional derivative in the direction of v at the given point. Remember to use a unit vector in your directional derivative computation. f(x, y) = ln(x + y), = 3i - 2j, P = (1,0)
Calculate the directional derivative in the direction of v at the given point. Remember to use a unit vector in your directional derivative computation. g (x, y, z) = z - xy + 2y, v = (1, -2, 2), P =
Calculate the directional derivative in the direction of v at the given point. Remember to use a unit vector in your directional derivative computation. g(x, y, z) = xe-y, v = (1, 1, 1), P = (1,2,0)
Calculate the directional derivative in the direction of v at the given point. Remember to use a unit vector in your directional derivative computation. g (x, y, z) = x ln(y+z), v= 2i -j+k, P=(2, e,
Find the directional derivative of ƒ (x, y) = x2 + 4y2 at the point P = (3, 2) in the direction pointing to the origin.
Find the directional derivative of ƒ(x, y, z) = xy + z3 at the point P = (3, −2, −1) in the direction pointing to the origin.
Determine the direction in which ƒ has maximum rate of increase from P, and give the rate of change in that direction. f(x,y) = xe", P = (2,0)
Determine the direction in which ƒ has maximum rate of increase from P, and give the rate of change in that direction. f(x, y) = x xy + y, P= (-1,4)
Determine the direction in which ƒ has maximum rate of increase from P, and give the rate of change in that direction. f(x, y, z) = P = (1, -1,3)
Determine the direction in which ƒ has maximum rate of increase from P, and give the rate of change in that direction. f(x, y, z) = xy z, P = (1,5,9)
Suppose that ∇ƒP = (2, −4, 4). Is ƒ increasing or decreasing at P in the direction v = (2, 1, 3)?
Let ƒ(x, y) = xex2−y and P = (1, 1). (a) Calculate ||Vfp||. (b) Find the rate of change of f in the direction Vfp. (c) Find the rate of change of f in the direction of a vector making an angle of
Let ƒ(x, y, z) = sin(xy + z) and P = (0, −1, π). Calculate Du ƒ(P), where u is a unit vector making an angle θ = 30° with ∇ƒP.
Let T(x, y) be the temperature at location (x, y) on a thin sheet of metal. Assume that ∇T = (y − 4, x + 2y). Let r(t) = (t2, t)be a path on the sheet. Find the values of t such that d dt T(r(t))
Find a vector normal to the surface x2 + y2 − z2 = 6 at P = (3, 1, 2).
Find a vector normal to the surface 3z3 + x2y − y2x = 1 at P = (1, −1, 1).
Find the two points on the ellipsoidwhere the tangent plane is normal to v = (1, 1, −2). 310 + + z = 1
Assume we have a local coordinate system at latitude L on the earth’s surface with east, north, and up as the x, y, and z directions, respectively. In this coordinate system, the earth’s angular
Use the geostrophic flow model to explain the following: In the Southern Hemisphere, winds blow with low pressure to the right, and the closer together the isobars, the stronger the winds. In
Use the Linear Approximation to ƒ(x, y) = √x/y at (9, 4) to estimate √9.1/3.9.
Use the Linear Approximation of ƒ(x, y) = ex2+y at (0, 0) to estimate ƒ(0.01, −0.02). Compare with the value obtained using a calculator.
Let ƒ(x, y) = x2/(y2 + 1). Use the Linear Approximation at an appropriate point (a, b) to estimate ƒ(4.01, 0.98).
Find the linearization of f (x, y, z) = z √x + y centered at (8, 4, 5).
Find the linearization of ƒ(x, y, z) = xy/z centered at (2, 1, 2). Use it to estimate ƒ(2.05, 0.9, 2.01) and compare with the value obtained from a calculator.
Estimate ƒ (2.1, 3.8) assuming that (2, 4) = 5, fx(2, 4) = 0.3, fy(2,4)= -0.2
Estimate ƒ(1.02, 0.01, −0.03) assuming that f(1,0,0) = -3, fy(1, 0, 0) = 4, fx(1,0,0) = -2 f(1,0,0) = 2
Use the Linear Approximation to estimate the value. Compare with the value given by a calculator.(2.01)3(1.02)2
Use the Linear Approximation to estimate the value. Compare with the value given by a calculator.4.1/7.9
Use the Linear Approximation to estimate the value. Compare with the value given by a calculator. 3.01 + 3.992
Use the Linear Approximation to estimate the value. Compare with the value given by a calculator. 0.98 2.013 + 1
Use the Linear Approximation to estimate the value. Compare with the value given by a calculator. (1.9)(2.02)(4.05)
Use the Linear Approximation to estimate the value. Compare with the value given by a calculator. 8.01 (1.99)(2.01)
Suppose that the plane tangent to z = ƒ(x, y) at (−2, 3, 4) has equation 4x + 2y + z = 2. Estimate ƒ(−2.1, 3.1).
The vector n = (2, −3, 6) is normal to the tangent plane to z = h(x, y) at (1, −3, 5). Estimate h(0.85, −3.08).
Let I = W/H2 denote the BMI described in Example 6.A child has weight W = 34 kg and height H = 1.3 m. Use the linear approximation to estimate the change in I if (W, H) changes to (36, 1.32) EXAMPLE
Let I = W/H2 denote the BMI described in Example 6.Suppose that (W, H) = (34, 1.3). Use the Linear Approximation to estimate the increase in H required to keep I constant if W increases to 35.
Let I = W/H2 denote the BMI described in Example 6.(a) Show that ΔI ≈ 0 if ΔH/ΔW ≈ H/2W.(b) Suppose that (W, H) = (25, 1.1). What increase in H will leave I (approximately) constant if W is
Let I = W/H2 denote the BMI described in Example 6.Estimate the change in height that will decrease I by 1 if (W, H) = (25, 1.1), assuming that W remains constant. EXAMPLE 6 Body Mass Index A

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