Calculus: Early Transcendentals, Loose-leaf Version, 9th
Calculus: Early Transcendentals, Loose-leaf Version, 9th
9th Edition
ISBN: 9780357022290
Author: Stewart
Publisher: Cengage Learning Acquisitions
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Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in ter…
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which …
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the …
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Chapter 14.7, Problem 24E

Show that f ( x , y )   = x 2 y e − x 2   −   y 2 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 of them give rise to maximum values? Minimum values? Saddle points?

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Chapter 14 Solutions

Calculus: Early Transcendentals, Loose-leaf Version, 9th

Ch. 14.1 - If f(x,y)=x2y/2xy2 , find (a) f(1,3) (b) f(2,1)...Ch. 14.1 - If g(x,y)=xsiny+ysinx , find (a) g(,0) (b)...Ch. 14.1 - Let F(x,y,z)=yx2z . (a) Evaluate F(3,4,1) . (b)...Ch. 14.1 - Let f(x,y,z)=lnzx2+y2 . (a) Evaluate f(4,3,6) ....Ch. 14.1 - Find and sketch the domain of the function. 7....Ch. 14.1 - Find and sketch the domain of the function. 8....Ch. 14.1 - Find and sketch the domain of the function. 9....Ch. 14.1 - Find and sketch the domain of the function. 10....Ch. 14.1 - Find and sketch the domain of the function. 11....Ch. 14.1 - Find and sketch the domain of the function. 12....
Ch. 14.1 - Find and sketch the domain of the function. 13....Ch. 14.1 - Find and sketch the domain of the function. 14....Ch. 14.1 - Find and sketch the domain of the function. 16....Ch. 14.1 - A model for the surface area of a human body is...Ch. 14.1 - Prob. 18ECh. 14.1 - The temperature-humidity index I (or humidex, for...Ch. 14.1 - A company makes three sizes of cardboard boxes:...Ch. 14.1 - Sketch the graph of the function. 23. f(x,y)=yCh. 14.1 - Sketch the graph of the function. 24. f(x,y)=x2Ch. 14.1 - Sketch the graph of the function. 25....Ch. 14.1 - Sketch the graph of the function. 26. f(x,y)=cosyCh. 14.1 - Sketch the graph of the function. 27. f(x,y)=sinxCh. 14.1 - Sketch the graph of the function. 28. f(x,y)=2x2y2Ch. 14.1 - Sketch the graph of the function. 29....Ch. 14.1 - Sketch the graph of the function. 30....Ch. 14.1 - Sketch the graph of the function. 31....Ch. 14.1 - Match the function with its graph (labeled IVI)....Ch. 14.1 - A contour map for a function f is shown. Use it to...Ch. 14.1 - Level curves (isothermals) are shown for the...Ch. 14.1 - Two contour maps are shown. One is for a function...Ch. 14.1 - Locate the points A and B on the map of Lonesome...Ch. 14.1 - Make a rough sketch of a contour map for the...Ch. 14.1 - The body mass index (BMI) of a person is defined...Ch. 14.1 - The body mass index is defined in Exercise 39....Ch. 14.1 - A contour map of a function is shown. Use it to...Ch. 14.1 - A contour map of a function is shown. Use it to...Ch. 14.1 - A contour map of a function is shown. Use it to...Ch. 14.1 - A contour map of a function is shown. Use it to...Ch. 14.1 - Draw a contour map of the function showing several...Ch. 14.1 - Draw a contour map of the function showing several...Ch. 14.1 - Draw a contour map of the function showing several...Ch. 14.1 - Draw a contour map of the function showing several...Ch. 14.1 - Draw a contour map of the function showing several...Ch. 14.1 - Draw a contour map of the function showing several...Ch. 14.1 - Draw a contour map of the function showing several...Ch. 14.1 - Draw a contour map of the function showing several...Ch. 14.1 - Sketch both a contour map and a graph of the...Ch. 14.1 - Sketch both a contour map and a graph of the...Ch. 14.1 - A thin metal plate, located in the xy-plane, has...Ch. 14.1 - If V(x, y) is the electric potential at a point...Ch. 14.1 - Graph the function using various domains and...Ch. 14.1 - Match the function (a) with its graph (labeled AF...Ch. 14.1 - Match the function (a) with its graph (labeled AF...Ch. 14.1 - Match the function (a) with its graph (labeled AF...Ch. 14.1 - Match the function (a) with its graph (labeled AF...Ch. 14.1 - Match the function (a) with its graph (labeled AF...Ch. 14.1 - Match the function (a) with its graph (labeled AF...Ch. 14.1 - Describe the level surfaces of the function. 68....Ch. 14.1 - Prob. 69ECh. 14.1 - Prob. 70ECh. 14.1 - Describe how the graph of g is obtained from the...Ch. 14.1 - Describe how the graph of g is obtained from the...Ch. 14.1 - Prob. 75ECh. 14.1 - Prob. 76ECh. 14.1 - Investigate the family of functions...Ch. 14.1 - Graph the functions f(x,y)=x2+y2 f(x,y)=ex2+y2...Ch. 14.2 - Prob. 1ECh. 14.2 - Explain why each function is continuous or...Ch. 14.2 - Prob. 3ECh. 14.2 - Prob. 4ECh. 14.2 - Prob. 5ECh. 14.2 - Prob. 6ECh. 14.2 - Prob. 7ECh. 14.2 - Prob. 8ECh. 14.2 - Prob. 9ECh. 14.2 - Prob. 10ECh. 14.2 - Prob. 11ECh. 14.2 - Find the limit. 12. lim(x,y)(,/2)cosysin2ycosxcosyCh. 14.2 - Prob. 13ECh. 14.2 - Prob. 14ECh. 14.2 - Prob. 15ECh. 14.2 - Prob. 16ECh. 14.2 - Show that the limit does not exist. 17....Ch. 14.2 - Prob. 18ECh. 14.2 - Prob. 19ECh. 14.2 - Find the limit, if it exists, or show that the...Ch. 14.2 - Prob. 21ECh. 14.2 - Prob. 22ECh. 14.2 - Prob. 23ECh. 14.2 - Prob. 24ECh. 14.2 - Prob. 25ECh. 14.2 - Prob. 26ECh. 14.2 - Find the limit, if it exists, or show that the...Ch. 14.2 - Prob. 28ECh. 14.2 - Prob. 29ECh. 14.2 - Prob. 30ECh. 14.2 - Prob. 31ECh. 14.2 - Prob. 32ECh. 14.2 - Prob. 33ECh. 14.2 - Prob. 34ECh. 14.2 - Prob. 35ECh. 14.2 - Prob. 36ECh. 14.2 - Prob. 37ECh. 14.2 - Find h(x, y) = g(f(x, y)) and the set of points at...Ch. 14.2 - Graph the function and observe where it is...Ch. 14.2 - Graph the function and observe where it is...Ch. 14.2 - Determine the set of points at which the function...Ch. 14.2 - Determine the set of points at which the function...Ch. 14.2 - Determine the set of points at which the function...Ch. 14.2 - Determine the set of points at which the function...Ch. 14.2 - Determine the set of points at which the function...Ch. 14.2 - Determine the set of points at which the function...Ch. 14.2 - Determine the set of points at which the function...Ch. 14.2 - Prob. 48ECh. 14.2 - Determine the set of points at which the function...Ch. 14.2 - Prob. 50ECh. 14.2 - Prob. 51ECh. 14.2 - Prob. 52ECh. 14.2 - Prob. 53ECh. 14.2 - Prove the three special limits in (2).Ch. 14.2 - At the beginning of this section we considered the...Ch. 14.2 - Graph and discuss the continuity of the function...Ch. 14.2 - Let f(x,y)={0ify0oryx41if0yx4 (a) Show that f(x,...Ch. 14.2 - Show that the function f given by f(x) = |x| is...Ch. 14.2 - If c Vn, show that the function f given by f(x) =...Ch. 14.3 - At the beginning of this section we discussed the...Ch. 14.3 - Prob. 2ECh. 14.3 - Prob. 4ECh. 14.3 - Prob. 5ECh. 14.3 - A contour map is given for a function f. Use it to...Ch. 14.3 - If f(x, y) = 16 4x2 y2, find fx(1, 2) and fy(1,...Ch. 14.3 - If f(x,y)=4x24y2, find fx(1,0) and fy(1, 0) and...Ch. 14.3 - Prob. 9ECh. 14.3 - Prob. 10ECh. 14.3 - Prob. 11ECh. 14.3 - Prob. 12ECh. 14.3 - Prob. 13ECh. 14.3 - Prob. 14ECh. 14.3 - Prob. 15ECh. 14.3 - Prob. 16ECh. 14.3 - Prob. 17ECh. 14.3 - Prob. 18ECh. 14.3 - Prob. 19ECh. 14.3 - Prob. 20ECh. 14.3 - Prob. 21ECh. 14.3 - Prob. 22ECh. 14.3 - Prob. 23ECh. 14.3 - Prob. 24ECh. 14.3 - Prob. 25ECh. 14.3 - Prob. 26ECh. 14.3 - Prob. 27ECh. 14.3 - Prob. 28ECh. 14.3 - Prob. 29ECh. 14.3 - Prob. 30ECh. 14.3 - Prob. 31ECh. 14.3 - Prob. 32ECh. 14.3 - Prob. 33ECh. 14.3 - Prob. 34ECh. 14.3 - Prob. 35ECh. 14.3 - Prob. 36ECh. 14.3 - Prob. 37ECh. 14.3 - Find the indicated partial derivative. 42....Ch. 14.3 - Find the indicated partial derivative. 43....Ch. 14.3 - Find the indicated partial derivative. 44. f(x, y,...Ch. 14.3 - Use implicit differentiation to find z/x and z/y ....Ch. 14.3 - Use implicit differentiation to find z/x and z/y ....Ch. 14.3 - Use implicit differentiation to find z/x and z/y ....Ch. 14.3 - Use implicit differentiation to find z/x and z/y ....Ch. 14.3 - Find z/x and z/y. 51. (a) z = f(x) + g(y) (b) z =...Ch. 14.3 - Find z/x and z/y. 52. (a) z = f(x)g(y) (b) z =...Ch. 14.3 - Prob. 47ECh. 14.3 - Find all the second partial derivatives. 54. f(x,...Ch. 14.3 - Find all the second partial derivatives. 55....Ch. 14.3 - Prob. 50ECh. 14.3 - Prob. 51ECh. 14.3 - Prob. 52ECh. 14.3 - Verify that the conclusion of Clairauts Theorem...Ch. 14.3 - Verify that the conclusion of Clairauts Theorem...Ch. 14.3 - Verify that the conclusion of Clairauts Theorem...Ch. 14.3 - Verify that the conclusion of Clairauts Theorem...Ch. 14.3 - Find the indicated partial derivative(s). 63. f(x,...Ch. 14.3 - Find the indicated partial derivative(s). 64. f(x,...Ch. 14.3 - Find the indicated partial derivative(s). 65....Ch. 14.3 - Prob. 60ECh. 14.3 - Find the indicated partial derivative(s). 67....Ch. 14.3 - Prob. 62ECh. 14.3 - Find the indicated partial derivative(s). 69....Ch. 14.3 - Find the indicated partial derivative(s). 70. u =...Ch. 14.3 - Use Definition 4 to find fx(x,y) and fy(x,y) . 65....Ch. 14.3 - Use Definition 4 to find fx(x,y) and fy(x,y) . 66....Ch. 14.3 - If f(x,y,z)=xy2z3+arcsin(xz), find fxzy. [Hint:...Ch. 14.3 - If g(x,y,z)=1+xz+1xy, find gxyz. [Hint: Which...Ch. 14.3 - Use the table of values of f(x, y) to estimate the...Ch. 14.3 - If u=ea1x1+a2x2++anxn, where a12+a22++an2=1, show...Ch. 14.3 - Show that the function u=u(x,t) is a solution of...Ch. 14.3 - Prob. 78ECh. 14.3 - Verify that the function u=1/x2+y2+z2 is a...Ch. 14.3 - Prob. 80ECh. 14.3 - The Diffusion Equation The diffusion equation...Ch. 14.3 - Prob. 82ECh. 14.3 - Prob. 83ECh. 14.3 - Prob. 84ECh. 14.3 - Show that the Cobb-Douglas production function...Ch. 14.3 - Prob. 86ECh. 14.3 - Prob. 87ECh. 14.3 - Prob. 88ECh. 14.3 - In the project following Section 4.7 we expressed...Ch. 14.3 - Prob. 91ECh. 14.3 - Prob. 92ECh. 14.3 - The ellipsoid 4x2+2y2+z2=16 intersects the plane...Ch. 14.3 - Prob. 95ECh. 14.3 - If a, b, c are the sides of a triangle and A, B, C...Ch. 14.3 - Prob. 97ECh. 14.3 - Prob. 98ECh. 14.3 - If f(x,y)=x(x2+y2)3/2esin(x2y) find fx(1,0)....Ch. 14.3 - If f(x,y)=x3+y33 find fx(0, 0).Ch. 14.3 - Prob. 1DPCh. 14.3 - Prob. 2DPCh. 14.4 - The graph of a function f is shown. Find an...Ch. 14.4 - The graph of a function f is shown. Find an...Ch. 14.4 - Find an equation of the tangent plane to the given...Ch. 14.4 - Find an equation of the tangent plane to the given...Ch. 14.4 - Find an equation of the tangent plane to the given...Ch. 14.4 - Find an equation of the tangent plane to the given...Ch. 14.4 - Find an equation of the tangent plane to the given...Ch. 14.4 - Find an equation of the tangent plane to the given...Ch. 14.4 - Find an equation of the tangent plane to the given...Ch. 14.4 - Find an equation of the tangent plane to the given...Ch. 14.4 - Graph the surface and the tangent plane at the...Ch. 14.4 - Graph the surface and the tangent plane at the...Ch. 14.4 - Explain why the function is differentiable at the...Ch. 14.4 - Explain why the function is differentiable at the...Ch. 14.4 - Explain why the function is differentiable at the...Ch. 14.4 - Explain why the function is differentiable at the...Ch. 14.4 - Explain why the function is differentiable at the...Ch. 14.4 - Explain why the function is differentiable at the...Ch. 14.4 - Explain why the function is differentiable at the...Ch. 14.4 - Explain why the function is differentiable at the...Ch. 14.4 - Verify the linear approximation at (0, 0). 17. ex...Ch. 14.4 - Verify the linear approximation at (0, 0). 18....Ch. 14.4 - Given that f is a differentiable function with...Ch. 14.4 - Find the linear approximation of the function f(x,...Ch. 14.4 - Find the linear approximation of the function...Ch. 14.4 - The wave heights h in the open sea depend on the...Ch. 14.4 - Use the table in Example 3 to find a linear...Ch. 14.4 - The wind-chill index W is the perceived...Ch. 14.4 - Find the differential of the function. 27. m =...Ch. 14.4 - Find the differential of the function. 35....Ch. 14.4 - Find the differential of the function. 25. z = e2x...Ch. 14.4 - Find the differential of the function. 26....Ch. 14.4 - Find the differential of the function. 35....Ch. 14.4 - Find the differential of the function. 36....Ch. 14.4 - Find the differential of the function. 29. R = 2...Ch. 14.4 - Find the differential of the function. 28....Ch. 14.4 - If z = 5x2 + y2 and (x, y) changes from (1, 2) to...Ch. 14.4 - If z = x2 xy + 3y2 and (x, y) changes from (3, 1)...Ch. 14.4 - The length and width of a rectangle are measured...Ch. 14.4 - Use differentials to estimate the amount of metal...Ch. 14.4 - Use differentials to estimate the amount of tin in...Ch. 14.4 - Prob. 44ECh. 14.4 - The radius of a right circular cylinder is...Ch. 14.4 - The wind-chill index is modeled by the function...Ch. 14.4 - Prob. 47ECh. 14.4 - The pressure, volume, and temperature of a mole of...Ch. 14.4 - If R is the total resistance of three resistors,...Ch. 14.4 - A model for the surface area of a human body is...Ch. 14.4 - Prob. 51ECh. 14.4 - Suppose you need to know an equation of the...Ch. 14.4 - Prove that if f is a function of two variables...Ch. 14.4 - (a) The function...Ch. 14.5 - Find dz/dt in two ways: by using the Chain Rule,...Ch. 14.5 - Find dz/dt in two ways: by using the Chain Rule,...Ch. 14.5 - Use the Chain Rule to find dz/dt or dw/dt. 1. z =...Ch. 14.5 - Use the Chain Rule to find dz/dt or dw/dt. 2....Ch. 14.5 - Use the Chain Rule to find dz/dt or dw/dt. 3. z =...Ch. 14.5 - Use the Chain Rule to find dz/dt or dw/dt. 4....Ch. 14.5 - Use the Chain Rule to find dz/dt or dw/dt. 5. w =...Ch. 14.5 - Use the Chain Rule to find dz/dt or dw/dt. 6. w =...Ch. 14.5 - Find z/s and z/t in two ways: by using the Chain...Ch. 14.5 - Use the Chain Rule to find z/s and z/t. 7. z = (x ...Ch. 14.5 - Use the Chain Rule to find z/s and z/t. 8. z =...Ch. 14.5 - Use the Chain Rule to find z/s and z/t. 9. z =...Ch. 14.5 - Prob. 14ECh. 14.5 - Prob. 16ECh. 14.5 - Suppose f is a differentiable function of x and y...Ch. 14.5 - Let R(s, t) = G(u(s, t), v(s, t)), where G, u, and...Ch. 14.5 - Suppose f is a differentiable function of x and y,...Ch. 14.5 - Suppose f is a differentiable function of x and y,...Ch. 14.5 - Prob. 21ECh. 14.5 - Prob. 22ECh. 14.5 - Prob. 23ECh. 14.5 - Prob. 24ECh. 14.5 - Use the Chain Rule to find the indicated partial...Ch. 14.5 - Use the Chain Rule to find the indicated partial...Ch. 14.5 - Use the Chain Rule to find the indicated partial...Ch. 14.5 - Use the Chain Rule to find the indicated partial...Ch. 14.5 - Use the Chain Rule to find the indicated partial...Ch. 14.5 - Use the Chain Rule to find the indicated partial...Ch. 14.5 - Use Equation 5 to find dy/dx . 31. ycosx=x2+y2Ch. 14.5 - Use Equation 5 to find dy/dx . 32. cos(xy)=1+sinyCh. 14.5 - Use Equation 5 to find dy/dx . 33. tan1x2y=x+xy2Ch. 14.5 - Use Equation 5 to find dy/dx . 34. eysinx=x+xyCh. 14.5 - Use Equations 6 to find z/x and z/y . 35....Ch. 14.5 - Prob. 36ECh. 14.5 - Prob. 37ECh. 14.5 - Prob. 38ECh. 14.5 - The temperature at a point (x, y) is T(x, y),...Ch. 14.5 - Prob. 40ECh. 14.5 - The speed of sound traveling through ocean water...Ch. 14.5 - Prob. 42ECh. 14.5 - Prob. 43ECh. 14.5 - The voltage V in a simple electrical circuit is...Ch. 14.5 - The pressure of 1 mole of an ideal gas is...Ch. 14.5 - A manufacturer has modeled its yearly production...Ch. 14.5 - One side of a triangle is increasing at a rate of...Ch. 14.5 - Prob. 48ECh. 14.5 - Assume that all the given functions are...Ch. 14.5 - Assume that all the given functions are...Ch. 14.5 - Prob. 51ECh. 14.5 - Prob. 52ECh. 14.5 - Prob. 53ECh. 14.5 - Prob. 54ECh. 14.5 - Prob. 55ECh. 14.5 - Prob. 56ECh. 14.5 - Homogeneous Functions A function f is called...Ch. 14.5 - Prob. 58ECh. 14.5 - Prob. 59ECh. 14.5 - Equation 5 is a formula for the derivative dy/dx...Ch. 14.6 - Level curves for barometric pressure (in...Ch. 14.6 - The contour map shows the average maximum...Ch. 14.6 - Find the directional derivative of f at the given...Ch. 14.6 - Find the directional derivative of f at the given...Ch. 14.6 - Find the directional derivative of f at the given...Ch. 14.6 - Find the directional derivative of f at the given...Ch. 14.6 - (a) Find the gradient of f . (b) Evaluate the...Ch. 14.6 - (a) Find the gradient of f. (b) Evaluate the...Ch. 14.6 - (a) Find the gradient of f. (b) Evaluate the...Ch. 14.6 - (a) Find the gradient of f. (b) Evaluate the...Ch. 14.6 - (a) Find the gradient of f. (b) Evaluate the...Ch. 14.6 - Find the directional derivative of the function at...Ch. 14.6 - Find the directional derivative of the function at...Ch. 14.6 - Find the directional derivative of the function at...Ch. 14.6 - Prob. 16ECh. 14.6 - Find the directional derivative of the function at...Ch. 14.6 - Find the directional derivative of the function at...Ch. 14.6 - Find the directional derivative of the function at...Ch. 14.6 - Use the figure to estimate Du, f(2, 2).Ch. 14.6 - Prob. 21ECh. 14.6 - Find the directional derivative of the function at...Ch. 14.6 - Find the maximum rate of change of f at the given...Ch. 14.6 - Find the maximum rate of change of f at the given...Ch. 14.6 - Find the maximum rate of change of f at the given...Ch. 14.6 - Find the maximum rate of change of f at the given...Ch. 14.6 - Find the maximum rate of change of f at the given...Ch. 14.6 - Find the maximum rate of change of f at the given...Ch. 14.6 - Find the directions in which the directional...Ch. 14.6 - Find all points at which the direction of fastest...Ch. 14.6 - Near a buoy, the depth of a lake at the point with...Ch. 14.6 - The temperature T in a metal ball is inversely...Ch. 14.6 - Prob. 38ECh. 14.6 - Suppose that over a certain region of space the...Ch. 14.6 - Suppose you are climbing a hill whose shape is...Ch. 14.6 - Prob. 41ECh. 14.6 - Shown is a topographic map of Blue River Pine...Ch. 14.6 - Show that the operation of taking the gradient of...Ch. 14.6 - Sketch the gradient vector f(4. 6) for the...Ch. 14.6 - Prob. 45ECh. 14.6 - Find equations of (a) the tangent plane and (b)...Ch. 14.6 - Find equations of (a) the tangent plane and (b)...Ch. 14.6 - Find equations of (a) the tangent plane and (b)...Ch. 14.6 - Find equations of (a) the tangent plane and (b)...Ch. 14.6 - Find equations of (a) the tangent plane and (b)...Ch. 14.6 - Find equations of (a) the tangent plane and (b)...Ch. 14.6 - Prob. 53ECh. 14.6 - Prob. 54ECh. 14.6 - If f(x, y) = xy, find the gradient vector f(3, 2)...Ch. 14.6 - If g(x, y) = x2 + y2 4x, find the gradient vector...Ch. 14.6 - Show that the equation of the tangent plane to the...Ch. 14.6 - Find the equation of the tangent plane to the...Ch. 14.6 - Prob. 59ECh. 14.6 - At what point on the ellipsoid x2 + y2 + 2z2 = 1...Ch. 14.6 - Are there any points on the hyperboloid x2 y2 z2...Ch. 14.6 - Show that the ellipsoid 3x2 + 2y2 + z2 = 9 and the...Ch. 14.6 - Show that every plane that is tangent to the cone...Ch. 14.6 - Show that every normal line to the sphere x2 + y2...Ch. 14.6 - Where does the normal line to the paraboloid z =...Ch. 14.6 - Prob. 66ECh. 14.6 - Show that the sum of the x-, y-, and z-intercepts...Ch. 14.6 - Prob. 68ECh. 14.6 - Find parametric equations tor the tangent line to...Ch. 14.6 - (a) The plane y + z = 3 intersects the cylinder x2...Ch. 14.6 - Where does the helix r(t) = cos t, sin t, t...Ch. 14.6 - Prob. 72ECh. 14.6 - Orthogonal Surfaces Two surfaces are called...Ch. 14.6 - Orthogonal Surfaces Two surfaces are called...Ch. 14.6 - Suppose that the directional derivatives of f(x,...Ch. 14.6 - Prob. 76ECh. 14.6 - Prob. 77ECh. 14.7 - Suppose (1, 1) is a critical point of a function f...Ch. 14.7 - Suppose (0, 2) is a critical point of a function y...Ch. 14.7 - Use the level curves in the figure to predict the...Ch. 14.7 - Use the level curves in the figure to predict the...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Prob. 14ECh. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Show that f(x, y) = x2 + 4y2 4xy + 2 has an...Ch. 14.7 - Show that f(x,y)=x2yex2y2 has maximum values at...Ch. 14.7 - Use a graph or level curves or both to estimate...Ch. 14.7 - Use a graph or level curves or both to estimate...Ch. 14.7 - Use a graph or level curves or both to estimate...Ch. 14.7 - Use a graph or level curves or both to estimate...Ch. 14.7 - Find the absolute maximum and minimum values of f...Ch. 14.7 - Find the absolute maximum and minimum values of f...Ch. 14.7 - Find the absolute maximum and minimum values of f...Ch. 14.7 - Find the absolute maximum and minimum values of f...Ch. 14.7 - Find the absolute maximum and minimum values of f...Ch. 14.7 - Find the absolute maximum and minimum values of f...Ch. 14.7 - Find the absolute maximum and minimum values of f...Ch. 14.7 - Find the absolute maximum and minimum values of f...Ch. 14.7 - For functions of one variable it is impossible for...Ch. 14.7 - If a function of one variable is continuous on an...Ch. 14.7 - Find the shortest distance from the |point (2, 0,...Ch. 14.7 - Find the point on the plane x 2y + 3z = 6 that is...Ch. 14.7 - Prob. 45ECh. 14.7 - Prob. 46ECh. 14.7 - Find three positive numbers whose sum is 100 and...Ch. 14.7 - Find three positive numbers whose sum is 12 and...Ch. 14.7 - Find the maximum volume of a rectangular box that...Ch. 14.7 - Find the dimensions of the box with volume 1000...Ch. 14.7 - Find the volume of the largest rectangular box in...Ch. 14.7 - Find the dimensions of the rectangular box with...Ch. 14.7 - Find the dimensions of a rectangular box of...Ch. 14.7 - The base of an aquarium with given volume V is...Ch. 14.7 - A cardboard box without a lid is to have a volume...Ch. 14.7 - A rectangular building is being designed to...Ch. 14.7 - If the length of the diagonal of a rectangular box...Ch. 14.7 - A model for the yield Y of an agricultural crop as...Ch. 14.7 - The Shannon index (sometimes called the...Ch. 14.7 - Three alleles (alternative versions of a gene) A,...Ch. 14.7 - Method of Least Squares Suppose that a scientist...Ch. 14.7 - Find an equation of the plane that passes through...Ch. 14.8 - Pictured are a contour map of f and a curve with...Ch. 14.8 - (a) Use a graphing calculator or computer to graph...Ch. 14.8 - Each of these extreme value problems has a...Ch. 14.8 - Prob. 4ECh. 14.8 - Each of these extreme value problems has a...Ch. 14.8 - Each of these extreme value problems has a...Ch. 14.8 - Each of these extreme value problems has a...Ch. 14.8 - Each of these extreme value problems has a...Ch. 14.8 - Each of these extreme value problems has a...Ch. 14.8 - Each of these extreme value problems has a...Ch. 14.8 - Each of these extreme value problems has a...Ch. 14.8 - Each of these extreme value problems has a...Ch. 14.8 - Prob. 16ECh. 14.8 - Prob. 17ECh. 14.8 - Prob. 18ECh. 14.8 - Prob. 19ECh. 14.8 - Use Lagrange multipliers to give an alternate...Ch. 14.8 - Find the extreme values of f on the region...Ch. 14.8 - Find the extreme values of f subject to both...Ch. 14.8 - Find the extreme values of f subject to both...Ch. 14.8 - Find the extreme values of f subject to both...Ch. 14.8 - Consider the problem of maximizing the function...Ch. 14.8 - Consider the problem of minimizing the function...Ch. 14.8 - Referring to Exercise 27, we now suppose that the...Ch. 14.8 - Use Lagrange multipliers to prove that the...Ch. 14.8 - Use Lagrange multipliers to give an alternate...Ch. 14.8 - Use Lagrange multipliers to give an alternate...Ch. 14.8 - Find the maximum and minimum volumes of a...Ch. 14.8 - The plane x + y + 2z = 2 intersects the paraboloid...Ch. 14.8 - The plane 4x 3y + 8z = 5 intersects the cone z2 =...Ch. 14.8 - (a) Maximize i=1nxiyi subject to the constraints...Ch. 14 - (a) What is a function of two variables? (b)...Ch. 14 - Prob. 2CCCh. 14 - Prob. 3CCCh. 14 - (a) What does it mean to say that f is continuous...Ch. 14 - Prob. 5CCCh. 14 - Prob. 6CCCh. 14 - Prob. 7CCCh. 14 - Prob. 8CCCh. 14 - Prob. 9CCCh. 14 - If z = f(x, y), what arc the differentials dx, dy,...Ch. 14 - Prob. 11CCCh. 14 - If z is defined implicitly as a function of x and...Ch. 14 - Prob. 13CCCh. 14 - Prob. 14CCCh. 14 - Prob. 15CCCh. 14 - Prob. 16CCCh. 14 - Prob. 17CCCh. 14 - Prob. 18CCCh. 14 - Prob. 19CCCh. 14 - Prob. 1TFQCh. 14 - Prob. 2TFQCh. 14 - Prob. 3TFQCh. 14 - Prob. 4TFQCh. 14 - Prob. 5TFQCh. 14 - Determine whether the statement is true or false....Ch. 14 - Prob. 7TFQCh. 14 - Prob. 8TFQCh. 14 - Determine whether the statement is true or false....Ch. 14 - Prob. 10TFQCh. 14 - Prob. 11TFQCh. 14 - Prob. 12TFQCh. 14 - Prob. 1ECh. 14 - Find and sketch the domain of the function. 2....Ch. 14 - Prob. 3ECh. 14 - Prob. 4ECh. 14 - Sketch several level curves of the function. 5....Ch. 14 - Sketch several level curves of the function. 6....Ch. 14 - Make a rough sketch of a contour map for the...Ch. 14 - The contour map of a function f is shown, (a)...Ch. 14 - Prob. 9ECh. 14 - Prob. 10ECh. 14 - Prob. 11ECh. 14 - Prob. 12ECh. 14 - Prob. 13ECh. 14 - Prob. 14ECh. 14 - Prob. 15ECh. 14 - Prob. 16ECh. 14 - Prob. 17ECh. 14 - The speed of sound traveling through ocean water...Ch. 14 - Prob. 19ECh. 14 - Prob. 20ECh. 14 - Prob. 21ECh. 14 - Prob. 22ECh. 14 - Prob. 23ECh. 14 - Prob. 24ECh. 14 - Find equations of (a) the tangent plane and (b)...Ch. 14 - Prob. 26ECh. 14 - Prob. 27ECh. 14 - Find equations of (a) the tangent plane and (b)...Ch. 14 - Find equations of (a) the tangent plane and (b)...Ch. 14 - Prob. 30ECh. 14 - Find the points on the hyperboloid x2 + 4y2 z2 =...Ch. 14 - Prob. 32ECh. 14 - Find the linear approximation of the function...Ch. 14 - The two legs of a right triangle are measured as 5...Ch. 14 - Prob. 35ECh. 14 - If v = x2sin y + yexy, where x = s + 2t and y =...Ch. 14 - Suppose z = f(x, y), where x = g(s, t), y = h(s,...Ch. 14 - Prob. 38ECh. 14 - Prob. 39ECh. 14 - Prob. 40ECh. 14 - Prob. 41ECh. 14 - If cos(xyz) = 1 + .x2y2 + z2, find zx and zy.Ch. 14 - Find the gradient of the function f(x,y,z)=x2eyz2.Ch. 14 - Prob. 44ECh. 14 - Prob. 45ECh. 14 - Find the directional derivative of f at the given...Ch. 14 - Prob. 47ECh. 14 - Prob. 48ECh. 14 - The contour map shows wind speed in knots during...Ch. 14 - Find parametric equations of the tangent line at...Ch. 14 - Prob. 51ECh. 14 - Prob. 52ECh. 14 - Prob. 53ECh. 14 - Prob. 54ECh. 14 - Find the absolute maximum and minimum values of f...Ch. 14 - Find the absolute maximum and minimum values of f...Ch. 14 - Prob. 57ECh. 14 - Use a graphing calculator or computer (or Newtons...Ch. 14 - Prob. 59ECh. 14 - Prob. 60ECh. 14 - Prob. 61ECh. 14 - Use Lagrange multipliers to find the maximum and...Ch. 14 - Prob. 63ECh. 14 - A pentagon is formed by placing an isosceles...Ch. 14 - A rectangle with length L and width W is cut into...Ch. 14 - Marine biologists have determined that when a...Ch. 14 - A long piece of galvanized sheet metal with width...Ch. 14 - For what values of the number r is the function...Ch. 14 - Suppose f is a differentiable function of one...Ch. 14 - If the ellipse x2/a2 + y2/b2 = 1 is to enclose the...Ch. 14 - Prob. 8PP
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Finding Local Maxima and Minima by Differentiation; Author: Professor Dave Explains;https://www.youtube.com/watch?v=pvLj1s7SOtk;License: Standard YouTube License, CC-BY