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mathematics
calculus

Questions and Answers of Calculus

Give definitions of continuity at a point and continuity on a set for a function of three variables.
Show that the function defined byAnd ( (0, 0, 0) = 0 is not continuous at (0, 0, 0).
Show that the function defined byAnd ( (0, 0, 0) = 0 is not continuous at (0, 0, 0).
In Problems 1-5, find the gradient ((. 1. ((x, y) = x2y + 3xy 2. ((x, y) = x3y - y3 3. ((x, y) = xexy 4. ((x, y) = x2y cos y 5. ((x, y) = x2y/(x + y)
In Problems 1-3, find the gradient vector of the given function at the given point p. then find the equation of the tangent plane at p.1. ((x, y) = x2y - xy2, p = (-2, 3)2. ((x, y) = x3y + 3xy2, p =
In Problems 1 and 12, find the equation w = T (x, y, z) of the tangent "hyperplane" at p. 1. ((x, y, z) = 3x2 - 2y2 + xz2, p = (1, 2, -1) 2. ((x, y, z) = xyz + x2, p = (2, 0, -3)
Find all points (x, y) at which the tangent plane to the graph of z = x2 - 6x + 2y2 - 10y + 2xy is horizontal.
Find all points (x, y) at which the tangent plane to the graph of z = x3 is horizontal.
Find parametric equations of the line tangent to the surface z = y2 + x3y at the point (2, 1, 9) whose projection on the xy-plane is (a) Parallel to the x-axis: (b) Parallel to the y-axis: (c)
Refer to Figure 1. Find the equation of the tangent plane to z = -10((xy( at (1, -1). Recall: d(x(/dx = (x( / x for x ( 0.
Mean Value Theorem for Several Variables If f is differentiable at each point of the line segment from a to b, then there exists on that line segment a point c between a and b such that((b) - ((a) =
Find all values of c that satisfy the Mean Value Theorem for Several Variables for the function ((x, y) = 9 - x2 - y2 where a = and b =
Find all values of c that satisfy the Mean Value Theorem for Several Variables for the function ((x, y) = (4 - x2 where a = and b =
Find the most general function ((p) satisfying (((p) = p.
Plot the graph of ((x, y) = -(xy( together with its gradient field.(a) Based on this and Figures 5 and 6, make a conjecture about the direction in which a gradient vector points.(b) Is f
Plot the graph of f (x, y) = sin x + sin y - sin(x + y) on 0 ( x ( 2(, 0 ( y ( 2(. Also draw the gradient field to see if your conjecture in Problem 29 (a) holds up.
Prove Theorem B for (a) The three-variable case and (b) The n-variable case. Denote the standard unit vectors by i1, i2, ( ( ( ( in.
In Problems 1-5, find the directional derivative of ( at the point p in the direction of a.1. ((x, y) = x2y; p = 1, 2); a = 3i - 4j2. ((x, y) = y2 ln x; p = 1, 4); a = i - j3. ((x, y) = 2x2 + xy -
In what direction u does ((x, y) = sin(3x - y decrease most rapidly at p = ((/6, (/4)?
Sketch the level curve of ((x, y) = y/x2 that goes through p = (1, 2). Calculate the gradient vector (((p) and draw this vector, placing its initial point at p. What should be true about (( (p)?
Follow the instructions of Problem 15 for ((x, y) = x2 + 4y2 and p = (2, 1).
Find the directional derivative of ((x, y) = e-x cos y at (0, (/3) in the direction toward the origin.
The temperature at (x, y, z) of a solid sphere centered at the origin is given by(a) By inspection, decide where the solid sphere is hottest. (b) Find a vector pointing in the direction of greatest
The temperature at (x, y, z) of a solid sphere centered at the origin is T(x, y, z) = 100e-(x2+y2=z2). It is hottest at the origin. Show that the direction of greatest decrease in temperature is
Find the gradient of ((x, y, z) = sin(x2 + y2 + z2. Show that the gradient always points directly toward the origin or directly away from the origin.
Suppose that the temperature T at the point (x, y, z) depends only on the distance from the origin. Show that the direction of greatest increase in T is either directly toward the origin or directly
The elevation of a mountain above sea level at the point (x, y) is ((x, y). A mountain climber at p notes that the slope in the easterly direction is -1/2 and the slope in the northerly direction is
The elevation of a mountain above sea level at (x, y) is 3000e-(x2+2y2)/100 meters. The positive x-axis points east and the positive y-axis points north. A climber is directly above (10, 10). If the
If the temperature of a plate at the point (x, y) is T(x, y) = 10 + x2 - y2, find the path a heat-seeking particle (which always moves in the direction of greatest increase in temperature) would
Do Problem 26 assuming that T(x, y) = 20 - 2x2 - y2?
The point P(1, -1, -10) is on the surface z = -10((xy(. Starting at P, in what direction u = u1i + u2j should one move in each case? (a) To climb most rapidly. (b) To stay at the same level. (c) To
The temperature T in degrees Celsius at (x, y, z) is given by T = 10/(x2 + y2 + z2). where distances are in meters. A bee is flying away from the hot spot at the origin on a spiral path so that its
Let u = (3i - 4j) /5 and v = (4i + 3j)/5 and suppose that at some point P, Du( = -6 and Dv( = -6 and Dv( = 17.(a) Find (( at P.(b) ((((((2 = (du()2 + (Dv()2 in part (a). Show that this relation
Figure 6 shows the contour map for a hill 60 feet high, which we assume has equation z = ((x, y).(a) A raindrop landing on the hill above point. A will reach the xy-plane at A( by following the path
According to Theorem A, the differentiability of ( at p implies the existence of Du((p) in all directions Show that the converse is false by consideringat the origin.
Plot the graph of z = x2 - y2 on -5 ( x ( 5, -5 ( y ( 5; also plot its contour map and gradient field, thus illustrating Theorems B and C. Then estimate the xy-coordinates of the point where a
Follow the directions of Problem 33 for z = x - x3/9 - y2 Plot the graph of z = x2 - y2 on -5 ( x ( 5, -5 ( y ( 5; also plot its contour map and gradient field, thus illustrating Theorems B and C.
In Problems 1-3, find a unit vector in the direction in which ( increases most rapidly at p. What is the rate of change in this direction?1. ((x, y) = x3 - y5; p = (2, -1)2. ((x, y) = ey sin x; p =
In Problems 1-3, find dw/dt by using the Chain Rule. Express your final answer in terms of t. 1. w = x2y3; x = t3, y = t2 2. w = x2y - y2x; x = cos t, y = sin t 3. w = ex sin y + ey sin x; x = 3t, y
The part of a tree normally sawed into lumber is the trunk, a solid shaped approximately like a right circular cylinder. If the radius of the trunk of a certain tree is growing 1/2 inch per year and
The temperature of a metal plate at (x, y) is e-x-3y degrees. A bug is walking northeast at a rate of (8 feet per minute (i.e., dx/dt = 2). From the bug's point of view, how is the temperature
A boy's toy boat slips from his grasp at the edge of a straight river. The stream carries it along at 5 feet per second. A crosswind blows it toward the opposite bank at 4 feet per second. If the boy
Sand is pouring onto a conical pile in such a way that at a certain instant the height is 100 inches and increasing at 3 inches per minute and the base radius is 40 inches and increasing at 2 inches
In Problems 1-3, use the method of Example 6( to find dy/dx. 1. x3 + 2x2y - y3 = 0 2. ye-x + 5x - 17 = 0 3. x sin y + y cos x = 0
If 3x2z + y3 - xyz3 = 0, find (z/(x?
If ye-x + z sin x = 0, find (x/(z.
Let z = ((x, y), where x = r cos ( and y = r sin (. Show that
The wave equation of physics is the partial differential equationwhere c is a constant. Show that if ( is any twice differentiable function then y(x, t) = 1/2[((x - ct) + ((x + ct)] satisfies this
Show that if w = ((r - s, s - t, t - r) then
Call a function ((x, y) homogeneous of degree 1 if ((tx, ty) = t((x, y) for all t > 0. For example, ((x, y) = x + yey/x satisfies this criterion Prove Euler's Theorem that such a function
Leaving from the same point P, airplane A flies due east while airplane B flies N 50° E. At a certain instant, A is 200 miles from P flying at 450 miles per hour, and B is 150 miles from P flying at
Recall Newton's Law of Gravitation, which asserts that the magnitude F of the force of attraction between objects of masses M and m is F = GMm/r2, where r is the distance between them and G is a
In Problems 1-3, find (w/(t by using the Chain Rule. Express your final answer in terms of s and t. 1. w = x2y; x = st, y = s - t 2. w = x2 - y ln x; x = s/t, y = s2t 3. w = ex2+y2; x = s sin t, y =
In Problems 1-3, find the equation of the tangent plane to the given surface at the indicated point.1. x2 + y2 + z2 = 16; (2, 3, (3)2. 8x2 + y2 + 8z2 = 16; (1, 2, (2/2)3. x2 - y2 + z2 + 1 = 0; (1, 3,
Find all points on the surface z = x2 - 2xy - y2 - 8x + 4y where the tangent plane is horizontal.
Find a point on the surface z = 2x2 + 3y2 where the tangent plane is parallel to the plane 8x - 3y - z = 0.
Show that the surfaces x2 + 4y + z2 = 0 and x2 + y2 + z2 - 6z + 7 = 0 are tangent to each other at (0, -1, 2); that is, show that they have the same tangent plane at (0, -1, 2).
Show that the surface z = x2 y and y = 1/4 x2 + 3/4 intersect at (1, 1, 1) and have perpendicular tangent planes there.
Find a point on the surface x2 + 2y2 + 3z2 = 12 where the tangent plane is perpendicular to the line with parametric equations: x = 1 + 2t, y = 3 + 8t, z = 2 -6t.
Show that the equation of the tangent plane to the ellipsoid x2 / (2 + y2 / b2 + z2 / c2 = 1 at (x0, y0, z0) can be written in the form x0x / (2 + y0y / b2 + z0z / c2 = 1
Find the parametric equations of the line that is tangent to the curve of intersection of the surfaces ((x, y, z) = 9x2 + 4y2 + 4z2 - 41 = 0 And g(x, y, z) = 2x2 - y2 + 3z2 - 10 = 10 at the point (1,
Find the parametric equations of the line that is tangent to the curve of intersection of the surfaces x = z2 and y = z3 at (1, 1, 1).
In determining the specific gravity of an object, its weight in air is found to be A = 36 pounds and its weight in water is W = 20 pounds, with a possible error in each measurement of 0.02 pound.
The period T of a pendulum of length L is given by T = 2((L/g, where g is the acceleration of gravity. Show that dT/T = 1/2 [dL / L - dg/g], and use this result to estimate the maximum percentage
The formula 1/R = 1/R1 + 1R2 determines the combined resistance R when resistors of resistance R1 and R2 are connected in parallel. Suppose that R1 and R2 were measured at 25 and 100 ohms,
A bee sat at the point (1, 2, 1) on the ellipsoid x2 + y2 + 2z2 = 6 (distances in feet). At t = 0, it took off along the normal line at a speed of 4 feet per second. Where and when did it hit the
Show that a plane tangent at any point of the surface xyz = k forms with the coordinate planes a tetrahedron of fixed volume and find this volume.
Find and simplify the equation of the tangent plane at (x0, y0, z0) to the surface (x + (y + (z = a. Then show that the sum of the intercepts of this plane with the coordinate axes is (2.
For the function ((x, y) = (x2 + y2, find the second order Taylor approximation based at (x0, y0) = (3, 4. Then estimate ((3.1, 3.9) using(a) The first order approximation,(b) The second order
For the function ((x, y) = tan((x2 + y2) / 64), find the second order Taylor approximation based at (x0, y0) = (0, 0). Then estimate ((0.2, -0.3) using(a) The first order approximation,(b) The second
In Problems 9-12, use the total differential dz to approximate the change in z as (x, y) moves from P to Q. Then use a calculator to find the corresponding exact change (z (to the accuracy of your
In Problems 1-10, find all critical points. Indicate whether each such point gives a local maximum or a local minimum, or whether it is a saddle point.1. ((x, y) = x2 + 4y2 -4x2. ((x, y) = x2 + 4y2 -
In Problems 1-2, find the global maximum value and global minimum value of ( on S and indicate where each occurs. 1. ((x, y) = 3x + 4y; S = {(x, y): 0 ( x ( 1, -1 ( y ( 1} 2. ((x, y) = x2 + y2; S =
Express a positive number N as a sum of three positive numbers such that the product of these three numbers is a maximum.
Use the methods of this section to find the shortest distance from the origin to the plane x + 2y + 3z = 12.
Find the dimensions of the closed rectangular box of volume V0 with minimum surface area.
Find the dimensions of the rectangular box of volume V0 for which the sum of the edge lengths is least.
A rectangular metal tank with open top is to hold 256 cubic feet of liquid. What are the dimensions of the tank that re-quire the least material to build?
A rectangular box, whose edges are parallel to the coordinate axes, is inscribed in the ellipsoid 96x2 + 4y2 + 4z2 = 36. What is the greatest possible volume for such a box?
Find the three-dimensional vector with length 9, the sum of whose components is a maximum.
Find the point on the plane 2x + 4y + 3z = 12 that is closest to the origin. What is the minimum distance?
Find the point on the paraboloid z = x2 + y2 that is closest to (1, 2.0). What is the minimum distance?
24. Find the minimum distance between the point (1, 2, 0) and the quadric cone z2 = x2 + y2.
An open gutter with cross section in the form of a trapezoid with equal base angles is to be made by bending up equal strips along both sides of a long piece of metal 12 inches wide. Find the base
Find the minimum distance between the lines having parametric equations x = t - 1, y = 2t, z = t + 3 and x = 3s, y = s + 2, z = 2s - 1.
Convince yourself that the maximum and minimum values of a linear function ((x, y) = ax + by + c over a closed polygonal set (i.e., a polygon and its interior) will always occur at a vertex of the
Use the result of Problem 27 to maximize 2x + y subject to the constraints 4x + y ( 8, 2x + 3y ( 14, x ( 0, and y ( 0. Begin by graphing the set determined by the constraints.
Find the maximum and minimum values of z = y2 - x2 on the closed triangle with vertices (0, 0), (1, 2), and (2, -2).
Find the least-squares line for the data (3, 2), (4, 3), (5, 4), (6, 4), and (7, 5).
Find the maximum and minimum values of z = 2x2 + y2 - 4x - 2y + S on the set bounded by the closed triangle with vertices (0, 0), (4, 0), and (0, 1).
Suppose that in Example 6, the costs were as follows: underwater $400/foot; along the bank $2001foot; and across land $300/foot. What path should be taken to minimize the cost and what is the minimum
Suppose that in Example 6, the costs were as follows: underwater $500/foot; along the bank $200/foot; and across land $100/foot. What path should be taken to minimize the cost and what is the minimum
Find the maximum and minimum values of ((x, y) = 10 + x + y on the disk x2 + y2 ( 9. Flint Parametrize the boundary by x = 3 cos t, y = 3 sin t, 0 s t s tar.
Find the maximum and mini mum values of ((x, y) = x2 + y2 on the ellipse with interior x2/a2 + y2/b2 ( 1, where a > b. Parametrize the boundary by x = a cos t, y = b sin t, 0 ( t ( 2r(.
A box is to be made where the material for the sides and the lid cost $0.25 per square foot and the cost for the bottom is $0.40 per square foot. Find the dimensions of a box with volume 2 cubic feet
A steel box without a lid having volume 60 cubic feet is to be made from material that costs $4 per square foot for the bottom and $1 per square foot for the sides. Welding the sides to the bottom
Suppose that the temperature T on the circular plate {(x, y}: x2 + y2 ( 1} is given by T = 2x2 + y2 - y. End the hottest and coldest spots on the plate.
Let (a, b, c) be a fixed point in the rust octant. Find the plane through this point that cuts off from the first octant the tetrahedron of minimum volume, and determine the resulting volume.
In Problems 1-3, let R= {(x, y): 1 ( x ( 4, 0 ( y ( 2}. Evaluate((x, y) dA, where ( is the given function. 1. 2. 3.
In Problems 1-3, sketch the solid whose volume is given by the following double integrals over the rectangle R = {(x, y): 0 ( x ( 2, 0 ( y ( 3}.1.2. 3.

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