All Matches

Solution Library

Expert Answer

Textbooks

Search Textbook questions, tutors and Books

Oops, something went wrong!

Change your search query and then try again

Result Not Found

Books FREE
Tutors
Study Help
Ask a Question
AI Study Help New

Search
Sign In
Register

study help
mathematics
precalculus

Questions and Answers of Precalculus

Find three positive numbers whose sum is 100 and whose product is a maximum.
Find the points on the surface y2 = 9 + xz that are closest to the origin.
Find the points on the cone z2 = x2 + y2 that are closest to the point (4, 2, 0).
Find the point on the plane x - 2y + 3z = 6 that is closest to the point (0, 1, 1).
Find the shortest distance from the point (2, 0, -3) to the plane x + y + z = 1.
If a function of one variable is continuous on an interval and has only one critical number, then a local maximum has to be an absolute maximum. But this is not true for functions of two variables.
For functions of one variable it is impossible for a continuous function to have two local maxima and no local minimum. But for functions of two variables such functions exist. Show that the
Find the absolute maximum and minimum values of f on the set D.f (x, y) = x3 - 3x - y3 + 12y, D is the quadrilateral whose vertices are (-2, 3), (2, 3), (2, 2), and (-2, -2)
Find the absolute maximum and minimum values of f on the set D.f (x, y) = 2x3 + y4, D = {(x, y) | x2 + y2 < 1}
Find the absolute maximum and minimum values of f on the set D.f (x, y) = xy2, D = {(x, y) | x > 0, y > 0, x2 + y2 < 3}
Find the absolute maximum and minimum values of f on the set D.f (x, y) = x + y - xy, D is the closed triangular region with vertices (0, 0), (0, 2), and (4, 0)
Find the absolute maximum and minimum values of f on the set D.f (x, y) = x2 + y2 - 2x, D is the closed triangular region with vertices (2, 0), (0, 2), and (0, -2)
Use a graphing device as in Example 4 (or Newton’s method or solve numerically using a calculator or computer) to find the critical points of f correct to three decimal places. Then classify the
Use a graphing device as in Example 4 (or Newton’s method or solve numerically using a calculator or computer) to find the critical points of f correct to three decimal places. Then classify the
Use a graphing device as in Example 4 (or Newton’s method or solve numerically using a calculator or computer) to find the critical points of f correct to three decimal places. Then classify the
Use a graphing device as in Example 4 (or Newton’s method or solve numerically using a calculator or computer) to find the critical points of f correct to three decimal places. Then classify the
Use a graph or level curves or both to estimate the local maximum and minimum values and saddle point(s) of the function. Then use calculus to find these values precisely. f(x, y) = sin x + sin y +
Use a graph or level curves or both to estimate the local maximum and minimum values and saddle point(s) of the function. Then use calculus to find these values precisely. f(x, y) = sin x + sin y +
Use a graph or level curves or both to estimate the local maximum and minimum values and saddle point(s) of the function. Then use calculus to find these values precisely.f (x, y) = (x - y)e-x2-y2
Use a graph or level curves or both to estimate the local maximum and minimum values and saddle point(s) of the function. Then use calculus to find these values precisely.f (x, y) = x2 + y2 + x-2y-2
Show that f (x, y) = x2ye-x2-y2 has maximum values at (±1, 1/√2) and minimum values at (±1, -1/√2). Show also that f has infinitely many other critical points and D = 0 at each of them. Which
Show that f (x, y) = x2 + 4y2 - 4xy + 2 has an infinite number of critical points and that D = 0 at each one. Then show that f has a local (and absolute) minimum at each critical point.
Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the
Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the
Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the
Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the
Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the
Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the
Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the
Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the
Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the
Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the
Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the
Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the
Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the
Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the
Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the
Use the level curves in the figure to predict the location of the critical points of f and whether f has a saddle point or a local maximum or minimum at each critical point. Explain your reasoning.
Use the level curves in the figure to predict the location of the critical points of f and whether f has a saddle point or a local maximum or minimum at each critical point. Explain your reasoning.
Suppose (0, 2) is a critical point of a function t with continuous second derivatives. In each case, what can you say about t?(a) txx(0, 2) = -1, gxy(0, 2) = 6, gyy(0, 2) = 1(b) txx(0, 2) = -1,
Suppose (1, 1) is a critical point of a function f with continuous second derivatives. In each case, what can you say about f ?(a) fxx(1, 1) = 4, fx y(1, 1) = 1, fyy(1, 1) = 2(b) fxx(1, 1) = 4,
Show that if z = f (x, y) is differentiable at x0 = (x0, y0), then f(x) – f(xo) – Vf(xo)· (x – xo) х — Хо lim х—Хо |x - xo|
Suppose that the directional derivatives of f (x, y) are known at a given point in two nonparallel directions given by unit vectors u and v. Is it possible to find ∇f at this point? If so, how
(a) Show that the function f (x, y) = 3√xy is continuous and the partial derivatives fx and fy exist at the origin but the directional derivatives in all other directions do not exist.(b) Graph f
(a) Two surfaces are called orthogonal at a point of intersection if their normal lines are perpendicular at that point. Show that surfaces with equations F(x, y, z) = 0 and G(x, y, z) = 0 are
(a) The plane y + z = 3 intersects the cylinder x2 + y2 = 5 in an ellipse. Find parametric equations for the tangent line to this ellipse at the point (1, 2, 1).(b) Graph the cylinder, the plane, and
Find parametric equations for the tangent line to the curve of intersection of the paraboloid z = x2 + y2 and the ellipsoid 4x2 + y2 + z2 = 9 at the point (-1, 1, 2).
Show that the pyramids cut off from the first octant by any tangent planes to the surface xyz = 1 at points in the first octant must all have the same volume.
Show that the sum of the x-, y-, and z-intercepts of any tangent plane to the surface √x + √y + √z = √c is a constant.
At what points does the normal line through the point (1, 2, 1) on the ellipsoid 4x2 + y2 + 4z2 = 12 intersect the sphere x2 + y2 + z2 = 102?
Where does the normal line to the paraboloid z = x2 + y2 at the point (1, 1, 2) intersect the paraboloid a second time?
Show that every normal line to the sphere x2 + y2 + z2 = r2 passes through the center of the sphere.
Show that every plane that is tangent to the cone x2 + y2 = z2 passes through the origin.
Show that the ellipsoid 3x2 + 2y2 + z2 = 9 and the sphere x2 + y2 + z2 - 8x - 6y - 8z + 24 = 0 are tangent to each other at the point (1, 1, 2). (This means that they have a common tangent plane at
Are there any points on the hyperboloid x2 - y2 - z2 = 1 where the tangent plane is parallel to the plane z = x + y?
At what point on the ellipsoid x2 + y2 + 2z2 = 1 is the tangent plane parallel to the plane x + 2y + z = 1?
Show that the equation of the tangent plane to the elliptic paraboloid z/c = x2/a2 + y2/b2 at the point (x0, y0, z0) can be written as2xx0/a2 + 2yy0/b2 + z+z0/c
Find the equation of the tangent plane to the hyperboloid x2/a2 + y2/b2 + z2/c2 = 1 at (x0, y0, z0) and express it in a form similar to the one in Exercise 51.
Show that the equation of the tangent plane to the ellipsoid x2/a2 + y2/b2 + z2/c2 = 1 at the point (x0, y0, z0) can be written asxx0/a2 + yy0/b2 + zz0/c2 = 1
If g(x, y) = x2 + y2 - 4x, find the gradient vector ∇g(1, 2) and use it to find the tangent line to the level curve g(x, y) = 1 at the point (1, 2). Sketch the level curve, the tangent line, and
If f(x, y) = xy, find the gradient vector ∇f(3, 2) and use it to find the tangent line to the level curve f (x, y) = 6 at the point (3, 2). Sketch the level curve, the tangent line, and the
Use a computer to graph the surface, the tangent plane, and the normal line on the same screen. Choose the domain carefully so that you avoid extraneous vertical planes. Choose the viewpoint so that
Use a computer to graph the surface, the tangent plane, and the normal line on the same screen. Choose the domain carefully so that you avoid extraneous vertical planes. Choose the viewpoint so that
Find equations of (a) the tangent plane and (b) the normal line to the given surface at the specified point. x4 + y4 + z4 = 3x2y2z2, (1, 1, 1)
Find equations of (a) the tangent plane and (b) the normal line to the given surface at the specified point. x + y + z = exyz, (0, 0, 1)
Find equations of (a) the tangent plane and (b) the normal line to the given surface at the specified point.xy + yz + zx = 5, (1, 2, 1)
Find equations of (a) the tangent plane and (b) the normal line to the given surface at the specified point.xy2z 3 = 8, (2, 2, 1)
Find equations of (a) the tangent plane and (b) the normal line to the given surface at the specified point.x = y2 + z2 + 1, (3, 1, -1)
Find equations of (a) the tangent plane and (b) the normal line to the given surface at the specified point.2(x - 2)2 + (y - 1)2 + (z - 3)2 = 10, (3, 3, 5)
(a) If u = (a, b) is a unit vector and f has continuous second partial derivatives, show that Du2 f = fxxa2 + 2fxyab + fyyb2(b) Find the second directional derivative of f (x, y) = xe2y in the
Sketch the gradient vector = f (4, 6) for the function f whose level curves are shown. Explain how you chose the direction and length of this vector. УА --5- (4, 6) --3- 4. 35 х 4 2.
Show that the operation of taking the gradient of a function has the given property. Assume that u and v are differentiable functions of x and y and that a, b are constants. (a) V(au + bv) = a Vu + b
Shown is a topographic map of Blue River Pine Provincial Park in British Columbia. Draw curves of steepest descent from point A (descending to Mud Lake) and from point B. Blue Rivero Blue iver Mud
Suppose you are climbing a hill whose shape is given by the equation z = 1000 - 0.005x2 - 0.01y2, where x, y, and z are measured in meters, and you are standing at a point with coordinates (60, 40,
The temperature at a point (x, y, z) is given byT(x, y, z) = 200e-x2-3y2-9z2where T is measured in 8C and x, y, z in meters.(a) Find the rate of change of temperature at the point P(2, -1, 2) in the
The temperature T in a metal ball is inversely proportional to the distance from the center of the ball, which we take to be the origin. The temperature at the point (1, 2, 2) is 120o.(a) Find the
Near a buoy, the depth of a lake at the point with coordinates (x, y) is z = 200 + 0.02x2 - 0.001y3, where x, y, and z are measured in meters. A fisherman in a small boat starts at the point (80, 60)
Find all points at which the direction of fastest change of the function f (x, y) = x2 + y2 - 2x - 4y is i + j.
Find the directions in which the directional derivative of f(x, y) = x2 + xy3 at the point (2, 1) has the value 2.
(a) Show that a differentiable function f decreases most rapidly at x in the direction opposite to the gradient vector, that is, in the direction of 2=f (x).(b) Use the result of part (a) to find the
Find the maximum rate of change of f at the given point and the direction in which it occurs.f (p, q, r) = arctan(pqr), (1, 2, 1)
Find the maximum rate of change of f at the given point and the direction in which it occurs.f (x, y, z) = x/(y + z), (8, 1, 3)
Find the maximum rate of change of f at the given point and the direction in which it occurs.f (x, y, z) = x ln(yz), (1, 2, 1/2)
Find the maximum rate of change of f at the given point and the direction in which it occurs.f (x, y) = sin(xy), (1, 0)
Find the maximum rate of change of f at the given point and the direction in which it occurs.f (s, t) = test, (0, 2)
Find the maximum rate of change of f at the given point and the direction in which it occurs.f (x, y) = 4y√x , (4, 1)
Find the directional derivative of f(x, y, z) = xy2z3 at P(2, 1, 1) in the direction of Q(0, -3, 5).
Find the directional derivative of f (x, y) = √xy at P(2, 8) in the direction of Q(5, 4).
Use the figure to estimate Du f (2, 2). ул (2, 2) Vf(2, 2) х
Find the directional derivative of the function at the given point in the direction of the vector v.h(r, s, t) = ln(3r + 6s + 9t), (1, 1, 1), v = 4i + 12j + 6k
Find the directional derivative of the function at the given point in the direction of the vector v.f(x, y, z) = xy2 tan-1z, (2, 1, 1), v = (1, 1, 1)
Find the directional derivative of the function at the given point in the direction of the vector v.f(x, y, z) = x2y + y2z, (1, 2, 3), v = (2, -1, 2)
Find the directional derivative of the function at the given point in the direction of the vector v.g(u, v) = u2e-v, (3, 0), v = 3i + 4j
Find the directional derivative of the function at the given point in the direction of the vector v.g(s, t) = √(t , s2, 4), v = 2i - j
Find the directional derivative of the function at the given point in the direction of the vector v.f (x, y) = x/x2 + y2 , (1, 2), v = (3, 5)
(a) Find the gradient of f .(b) Evaluate the gradient at the point P.(c) Find the rate of change of f at P in the direction of the vector u. f(x, y, z) = y²eye, P(0, 1, – 1), u= (13 13 13/ xyz
(a) Find the gradient of f .(b) Evaluate the gradient at the point P.(c) Find the rate of change of f at P in the direction of the vector u. (0. §. –}) 3 f(x, y, z) = x²yz – xyz², P(2, – 1,
(a) Find the gradient of f .(b) Evaluate the gradient at the point P.(c) Find the rate of change of f at P in the direction of the vector u. f(x, y) = x² In y, P(3, 1), 12 13 13 J u =
(a) Find the gradient of f .(b) Evaluate the gradient at the point P.(c) Find the rate of change of f at P in the direction of the vector u. и 3D+ f (x, у) — х/у, Р(2, 1),
A table of values for the wind-chill index W = f (T, v) is given in Exercise 14.3.3 on page 923. Use the table to estimate the value of Du f (-20, 30), where u = (i + j)/√2.

Showing 18200 - 18300 of 29459

First
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
Last

Services

Sitemap
Fun
Definitions
Become Tutor
Study Help Categories
Recent Questions
Expert Questions

Company Info

Security
Copyrights
Privacy Policy
Terms & Conditions
SolutionInn Fee
Scholarship

Get In Touch

About Us
Contact Us
Career
Jobs
FAQ
Campus Ambassador

Secure Payment

Download Our App

© 2024 SolutionInn. All Rights Reserved