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mathematics
calculus

Questions and Answers of Calculus

How close does the great-circle route from St. Paul to Turin get to the North Pole?
Change the following from spherical to Cartesian coordinates. a. (8, π/4, π/6) b. (4, π/3, 3π/4)
As you may have guessed, there is a simple formula for expressing great-circle distance directly in terms of longitude and latitude. Let (α1, β1) and (α2, β2) be the longitude-latitude
Use Problem 40 to find the great-circle distance between each pair of places. a. New York and Greenwich b. St. Paul and Turin c. Turin and the South Pole (use al = a2) d. New York and Cape Town
It is easy to see that the graph of ( = 2a cos ϕ is a sphere of radius a sitting on the xy-plane at the origin. But what is the graph of ( = 2a sin ϕ?
Change the following from Cartesian to spherical coordinates. a. (2, - 2√3, 4) b. (- √2, √2, 2√3)
Change the following from Cartesian to cylindrical coordinates. a. (2, 2, 3) b. (4√3, -4, 6)
In Problems a-c, sketch the graph of the given cylindrical or spherical equation. a. r = 5 b. ( = 5 c. ϕ = π/6
In Problems 1-3, sketch a graph of the given function. 1. ((x, y) = (64 - x2 - y2 2. ((x, y) = 9 - x2 - y2 3. ((x, y) = x2 + 4y2
Evaluate the integrals in Problems 1-3.1.2. 3.
The solid in three-space consisting of those points whose spherical coordinates satisfy ( ( 7.
The solid in three-space bounded above by z = 9 - x2 - y2 and below by the xy-plane. Interpret this as a solid of revolution.
In Problems 1-3, sketch the graph of the given cylindrical or spherical equation. 1. r = 2 2. ( = 2 3. ( = (/4
Find the minimum of ((x, y) = x2 + y2 subject to the constraint g(x, y) = xy - 3 = 0.
Find the minimum distance between the origin and the surface x2y - z2 + 9 = 0.
Find the maximum volume of a closed rectangular box with faces parallel to the coordinate planes inscribed in the ellipsoidx2 / (2 + y2 / b2 + z2 / c2 = 1
Find the maximum volume of the first-octant rectangular box with faces parallel to the coordinate planes, one vertex at (0, 0, 0), and diagonally opposite vertex on the planex/( + y/b + z/c = 1
In Problems 1-4, use the method of Lagrange multipliers to solve these problems from section 12.81. ((x, y) = 10 + x + y; S = {(x, y): x2 + y2 ( 1}2. ((x, y) = x + y - xy; S = {(x, y): x2 + y2 ( 9}3.
Find the maximum of ((x. y) = xy subject to the constraint g(x, y) = 4x2 + 9y2 - 36 = 0.
In Problems 1-2, find the maximum and minimum of the friction f over the closed and bounded set S. Use the methods of Section 12.8 to find the maximum and minimum on the interior of S; then use
Find the shape of the triangle of maximum perimeter that can be inscribed in a circle of radius r. Let (, (, and ( be as in Figure 7 and reduce the problem to maximizing P = 2r(sin (/2 + sin (/2 +
Consider the Cobb-Douglas production model for a manufacturing process depending on three inputs x, y, and z with unit costs a, b, and c, respectively, given byP = kx(y(z(, ( > 0, ( > 0, ( >
Find the minimum distance from the origin to the line of intersection of the two planesx + y + z = 8 and 2x - y + 3z = 28
Find the maximum and minimum of ((x, y, z) = -x + 2y + 2z on the ellipse x2 + y2 = 2, y + 2z = 1.
Find the maximum of ((x, y) = 4x2 - 4xy + y2 subject to the constraint x2 + y2 = 1.
Let w = x1x2 ( ( ( +xn.(a) Maximize w subject to x1 + x2 + ( ( ( + xn = 1 and all xi > 0.(b) Use part (a) to deduce the famous Geometric Mean Arithmetic Mean Inequality for positive numbers a1,
Maximize w = (1x1 + (2x2 + ( ( ( + (nxn, all (i > 0, subject to x12 + x22 + ( ( ( + xn2 = 1.
Find the minimum of ((x, y, z) = x2 + y2 + z2 subject to the constraint x + 3y - 2z = 12.
Find the minimum of ((x, y, z) = 4x - 2y + 3z subject to the constraint 2x2 + y2 - 3z = 0.
What are the dimensions of the rectangular box, open at the top, that has maximum volume when the surface area is 48?
Find the minimum distance between the origin and the plane x + 3y - 2z = 4.
The material for the bottom of a rectangular box costs three times as much per square foot as the material for the sides and top. Find the greatest volume that such a box can have if the total amount
Find and sketch the domain of each indicated function of two variables, showing clearly any points on the boundary of the domain that belong to the domain. (a) z = (x2 + 4y2 - 100 (b) z = -(2x - y - 1
In each case, find the indicated limit or state that it does not exist.(a)(b) (c)
Find (((1, 2, -1).(a) ((x, y, z) = x2 yz3(b) ((x, y, z) = y2 sin xz
Find the directional derivative of ((x, y) = tan-1(3xy). What is its value at the point (4, 2) in the direction u = ((3/2)i - (1/2)j?
Find the slope of the tangent line to the curve of intersection of the vertical plane x - (3y + 2(3 - 1 = 0 and the surface z - x2 + y2 at the point (1, 2, 5).
For ((x, y) = x2/2 + y2,(a) Find the equation of its level curve that goes through the point (4, 1) in its domain;(b) Find the gradient vector (( at (4, 1);(c) Draw the level curve and draw the
If F(u, v) = tan-1(uv), u = (xy, and v = (x - (y, find (F/(x and (F/(y in terms of u, v, x, and y.
If ((u, v) = u/v, u = x2 - 3y + 4z, and v = xyz, find (x, (y, and (z in terms of x, y, and z.
Sketch the level curves of ((x, y) = (x + y2) for k = 0, 1, 2, 4.
If F(x, y) = x3 - xy2 - y4, x = 2 cos3t, and y = 3 sin t, find dF/dt at t = 0.
If F(x, y, z) = (5x2y/z3), x = t3/2 + 2, y = ln 4t, and z = e3t, find dF/ dt in terms of x. y, z, and t.
A triangle has vertices A, B, and C. The length of the side c = AB is increasing at the rate of 3 inches per second, the side b = AC is decreasing at 1 inch per second, and the included angle ( is
Find the gradient vector of F(x, y, z) = 9x2 + 4y2 + 9z2 - 34 at the point P(1, 2, -1). Write the equation of the tangent plane to the surface F(x, y, z) = 0 at P.
A right circular cylinder is measured to have a radius of 10 ± 0.02 inches and a height of 6 ± 0.01 inches. Calculate its volume and use differentials to give an estimate of the possible error.
If ((x, y, z) = xy2 / (1 + z2), use differentials to estimate ((1.01, 1.98, 2.03).
Find the extrema of ((x, y) = x2y - 6y2 - 3x2.
A rectangular box whose edges are parallel to the coordinate axes is inscribed in the ellipsoid 36x2 + 4y2 + 9z2 = 36. What is the greatest possible volume for such a box?
Use Lagrange multipliers to find the maximum and the minimum of ((x, y) = xy subject to the constraint x2 + y2 = 1.
Use Lagrange multipliers to find the dimensions of the right circular cylinder with maximum volume if its surface area is 24(.
In Problems 1-2, find (( / (x, (2( / (x2, and (2(/(y (x.1. ((x, y) = 3x4y2 + 7x2y72. ((x, y) = cos2 x - sin2 y
If ( is the function of three variables defined by ((x, y, z) xy3 - 5x2yz4, find (x(2, -1, 1), (y(2, -1, 1), and (z(2, -1, 1).
Let ((x, y) = x2y + (y. Find each value.(a) ((2, 1)(b) ((3, 0)(c) ((1, 4)(d) (((, (4)(e) ((1/x, x4)(f) ((2, -4)What is the natural domain for this function?
In Problems 1-3 sketch the level curve z = k for the indicated values of k.1.2. 3.
Let ((x, y) = y/x + xy. Find each value.(a) ((1, 2)(b) ((1/4, 4)(c) ((4, 1/4)(d) (((, ()(e) ((1/x, x2)(f) ((0, 0)What is the natural domain for this function?
Let T(x, y) be the temperature at a point (x, y) in the plane. Draw the isothermal curves corresponding to T = 0 if
If V(x, y) is the voltage at a point (x, y) in the plane, the level curves of V are called equipotential curves. Draw the equipotential curves corresponding to V = 1/2, 1, 2, 4 for
Figure 20 shows isotherms for the United States. (a) Which of San Francisco, Denver, and New York had approx-imately the same temperature as St. Louis? (b) If you were in Kansas City and wanted to
Figure 23 shows a contour map for barometric pressure in millibars. Level curves for barometric pressure are called isobars.(a) What part of the country had the lowest barometric pres-sure? The
In Problems 1-3 describe geometrically the domain of each of the indicated functions of three variables.1.2. ((x, y, z) = ln(x2 + y2 z2)3. ((x, y, z) = z ln(xy)
Let g(x, y, z) = x2 sin yz. Find each value.(a) g(1, (, 2)(b) g(2, 1, (/6)(c) g(4, 2, (/4)(d) g((, (, ()
Describe geometrically the level surfaces for the function defined in Problems 1-3. 1. ((x, y, z) = x2 + y2 + z2; k > 0 2. ((x, y, z) = 100x2 + 16y2 + 25z2; k > 0 3. ((x, y, z) = 16x2 + 16y2 - 9z2
Find the domain of each function.(a)(b) (c)
Let g(x, y, z) = (x cosy + z2. Find each value. (a) g(4, 0, 2) (b) g( -9, (, 3) (c) g(2, (/3, -1) (d) g(3, 6, 1.2)
Sketch (as best you can) the graph of the monkey saddle z = x(x2 - 3y2). Begin by nothing where z = 0.
The contour map in Figure 24 shows level curves for a mountain 3000 feet high.(a) What is special about the path to the top labeled AC? What is special about BC?(b) Make good estimates of the total
For each of the functions in Problems 1-3, draw the graph and the corresponding contour plot.1. ((x y) = sin(2x2 + y2; -2 ( x ( 2, -2 ( y ( 22. ((x, y) = sin(x2 + y2)/(x2 + y2), ((0, 0) = 1; -2 ( x
In Problems 1-3, sketch the graph off. 1. ((x, y) = 6 2. ((x, y) = 6 - x 3. ((x, y) = 6 - x - 2y
In Problems 1-3, find all first partial derivatives of each function.1. ((x, y) = (2x - y)42. ((x, y) = (2x - y2)3/23.
In Problems 1-3, verify that1. ((x, y) = 2x2y3 - x3 y5 2. ((x, y) = x3 + y2)5 3. ((x, y) = 3e2x cos y
If F(x, y) = 2x - y / xy, find Fx(3, - 2) and Fy(3, - 2).
If F(x, y) = ln(x2 + xy + y2), find Fx(-1, 4) and Fy(-1, 4).
If ((x, y) = tan-1(y2/x), find (x((5, -2) and (y((5, -2).
Find the slope of the tangent to the curve of intersection of the surface 2z = (36 - 9x2 - 4y2 and the plane x = 1 at the point (1, -2, (11/3).
Find the slope of the tangent to the curve of intersection of the cylinder 4z = 5(16 - x2 and the plane y = 3 at the point (2, 3, 5(3/2).
Show that, for the gas law of Problem 31,And According to the ideal gas law, the pressure, temperature, and volume of a gas are related by PV = kt, where k is a constant. Find the rate of change of
A function of two variables that satisfies Laplace's Equation,Is said be harmonic. Show that the functions defined in Problems 1 and 2 are harmonic functions.1. ((x, y) = x3 y - xy32. ((x, y) =
If ((x, y) = cos(2x2 - y2), find (3((x, y) /(y (x2.
Express the following in ( notation. (a) (yyy (b) (xxy (c) (xyyy
If ((x, y, z) = 3x2y - xyz + y2z2, find each of the following:(a) (x(x, y, z)(b) (y(0, 1, 2)(c) (xy(x, y, z)
If ((x, y, z) = (x3 + y2 + z)4, find each of the following:(a) (x(x, y, z)(b) (y(0, 1, 1)(c) (zz(x, y, z)
A bee was flying upward along the curve that is the inter-section of z = x4 xy3 + 12 with the plane x = 1. At the point (1, -2, 5), it went off on the tangent line. Where did the bee hit the xz-plane?
Let A(x, y) be the area of a nondegenemte rectangle of dimensions x and y, the rectangle being inside a circle of radius 10. Determine the domain and range for this function.
The interval [0, 1] is to be separated into three pieces by making cuts at x and y. Let A(x, y) be the area of any nondegen-crate triangle that can be formed from these three pieces. Deter-mine the
The wave equation c2 (2u/(x2 = (2u/(t2 and the heat equation c (2u/(x2 = (u/(t are two of the most important equations in physics (c is a constant). These are called partial differential equations.
For the contour map for z = ((x, y) shown in Figure 4, estimate each value.(a) (y(1,1)(b) (x(-4, 2)(c) (x(-5, -2)(d) (y(0, -2)
A CAS can be used to calculate and graph partial derivatives. Draw the graphs of each of the following: (a) sin(x + y2) (b) Dx sin(x + y2) (c) Dy sin(x + y2) (d) Dx(Dy sin(x + y2))
Give definitions in terms of limits for the following partial derivatives:(a) (y(x, y, z)(b) (z(x, y, z)(c) Gx(w, x, y, z)(d) (/(z ((x, y, z, t)(e) (/(b2 S(b0, b1, b2, ( ( ( ( bn)
Find each partial derivative. (a) (/(w (sin w sin x cos y cos z) (b) (/(x [x ln (wxyz)] (c) (t(x, y, z, t), where ((x, y, z, t) = t cos x / 1 + xyzt
In Problems 1-3, find the indicated limit or state that it does not exist.1.2. 3.
In Problems 1-3, describe the largest set S on which it is correct to say that ( is continuous.1.2. ((x, y) = ln(1 + x2 + y2)3. ((x, y) = ln(1 - x2 - y2)
In Problems 1-3, sketch the indicated set. Describe the boundary of the set. Finally, state whether the set in open, closed, or neither. 1. {(x, y): 2 ( x ( 4, 1 ( y ( 5} 2. {(x, y): x2 + y2 < 5} 3.
Let H be the hemispherical shell x2 + y2 + (z - 1)2 = 1, 0 ( z (a) ((x, y, z) is the time required for a particle dropped from (x, y, z) to reach the level z = 0.(b) ((x, y, z) is the area of the
Let (, a function of n variable, be continuous on an open set D, and suppose that P0 is in D, with ( (P0) > 0. Prove that there is a ( > 0 such that ( (P) > 0 in a neighborhood of P0 with
The French Railroad Suppose that Paris is located at the origin of the xy-plane. Rail lines emanate from Paris along all rays, and these are the only rail lines. Determine the set of discontinuities
Let ((x, y) = xy x2 - y2 / x2 + y2 if (x, y) ( (0, 0) and ((0, 0) = 0.Show that (xy(0, 0) ( (yx(0, 0) by completing the following steps:(a) Show thatFor all y.(b) Similarly, show that (y(x, 0) = x
Plot the graph of the function mentioned in Problem 42. Do you see why this surface is sometimes called the dog saddle?
Plot the graphs of each of the following functions on -2 ( x ( 2, -2 ( y ( 2, and determine where on this set they are discontinuous. (a) ((x, y) = x2 / (x2 + y2), ((0, 0) = 0 (b) ((x, y) = tan(x2 +
Plot the graph of ((x, y) = x2y/(x4 + y2) in an orientation that illustrates its unusual characteristics.

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