Bundle: Calculus, Loose-Leaf Version, 8th + WebAssign Printed Access Card for Stewart's Calculus, 8th Edition, Multi-Term
8th Edition
ISBN: 9781305616684
Author: James Stewart
Publisher: Cengage Learning
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Textbook Question
Chapter 14.5, Problem 37E
The speed of sound traveling through ocean water with salinity 35 parts per thousand has been modeled by the equation
where C is the speed of sound (in meters per second), T is the temperature (in degrees Celsius), and D is the depth below the ocean surface (in meters). A scuba diver began a leisurely dive into the ocean water; the diver’s depth and the surrounding water temperature over time are recorded in the following graphs. Estimate the rate of change (with respect to time) of the speed of sound through the ocean water experienced by the diver 20 minutes into the dive. What are the units?
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Students have asked these similar questions
Given lim x-4 f (x) = 1,limx-49 (x) = 10, and lim→-4 h (x) = -7 use the limit properties
to find lim→-4
1
[2h (x) — h(x) + 7 f(x)] :
-
h(x)+7f(x)
3
O DNE
17. Suppose we know that the graph below is the graph of a solution to dy/dt = f(t).
(a) How much of the slope field can
you sketch from this information?
[Hint: Note that the differential
equation depends only on t.]
(b) What can you say about the solu-
tion with y(0) = 2? (For example,
can you sketch the graph of this so-
lution?)
y(0) = 1
y
AN
(b) Find the (instantaneous) rate of change of y at x = 5.
In the previous part, we found the average rate of change for several intervals of decreasing size starting at x = 5. The instantaneous rate of
change of fat x = 5 is the limit of the average rate of change over the interval [x, x + h] as h approaches 0. This is given by the derivative in the
following limit.
lim
h→0
-
f(x + h) − f(x)
h
The first step to find this limit is to compute f(x + h). Recall that this means replacing the input variable x with the expression x + h in the rule
defining f.
f(x + h) = (x + h)² - 5(x+ h)
=
2xh+h2_
x² + 2xh + h² 5✔
-
5
)x - 5h
Step 4
-
The second step for finding the derivative of fat x is to find the difference f(x + h) − f(x).
-
f(x + h) f(x) =
= (x²
x² + 2xh + h² -
])-
=
2x
+ h² - 5h
])x-5h) - (x² - 5x)
=
]) (2x + h - 5)
Macbook Pro
Chapter 14 Solutions
Bundle: Calculus, Loose-Leaf Version, 8th + WebAssign Printed Access Card for Stewart's Calculus, 8th Edition, Multi-Term
Ch. 14.1 - In Example 2 we considered the function W=f(T,v),...Ch. 14.1 - The temperature-humidity index I or humidex, for...Ch. 14.1 - Prob. 3ECh. 14.1 - Verify for the Cobb-Douglas production function...Ch. 14.1 - Prob. 5ECh. 14.1 - Prob. 6ECh. 14.1 - Prob. 7ECh. 14.1 - A company makes three sizes of cardboard boxes:...Ch. 14.1 - Prob. 9ECh. 14.1 - Let F(x,y)=1+4y2. aEvaluate F(3,1). bFind and...
Ch. 14.1 - Let f(x,y,z)=x+y+z+ln(4x2y2z2). aEvaluate...Ch. 14.1 - Let g(x,y,z)=x3y2z10xyz. aEvaluate g(1,2,3). bFind...Ch. 14.1 - Find and sketch the domain of the function....Ch. 14.1 - Find and sketch the domain of the function....Ch. 14.1 - Prob. 15ECh. 14.1 - Prob. 16ECh. 14.1 - Find and sketch the domain of the function....Ch. 14.1 - Find and sketch the domain of the function....Ch. 14.1 - Find and sketch the domain of the function....Ch. 14.1 - Prob. 20ECh. 14.1 - Find and sketch the domain of the function....Ch. 14.1 - Prob. 22ECh. 14.1 - Prob. 23ECh. 14.1 - Prob. 24ECh. 14.1 - Prob. 25ECh. 14.1 - Prob. 26ECh. 14.1 - Sketch the graph of the function. f(x,y)=sinxCh. 14.1 - Prob. 28ECh. 14.1 - Prob. 29ECh. 14.1 - Sketch the graph of the function. f(x,y)=4x2+y2Ch. 14.1 - Sketch the graph of the function. f(x,y)=44x2y2Ch. 14.1 - Match the function with its graph labeled IVI....Ch. 14.1 - Prob. 33ECh. 14.1 - Shown is a contour map of atmospheric pressure in...Ch. 14.1 - Level curves isothermals are shown for the typical...Ch. 14.1 - Two contour maps are shown. One is for a function...Ch. 14.1 - Prob. 37ECh. 14.1 - Prob. 38ECh. 14.1 - The body mass index BMI of a person is defined by...Ch. 14.1 - Prob. 40ECh. 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 - Prob. 45ECh. 14.1 - Prob. 46ECh. 14.1 - Prob. 47ECh. 14.1 - Draw a contour map of the function showing several...Ch. 14.1 - Prob. 49ECh. 14.1 - Draw a contour map of the function showing several...Ch. 14.1 - Prob. 51ECh. 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 - Prob. 54ECh. 14.1 - Prob. 55ECh. 14.1 - Prob. 56ECh. 14.1 - Prob. 57ECh. 14.1 - Use a computer to graph the function using various...Ch. 14.1 - Prob. 59ECh. 14.1 - Prob. 60ECh. 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 - Prob. 67ECh. 14.1 - Prob. 68ECh. 14.1 - Prob. 69ECh. 14.1 - Describe the level surfaces of the function....Ch. 14.1 - Prob. 71ECh. 14.1 - Describe how the graph of g is obtained from the...Ch. 14.1 - Prob. 73ECh. 14.1 - Prob. 74ECh. 14.1 - Graph the function using various domains and...Ch. 14.1 - Prob. 76ECh. 14.1 - Prob. 77ECh. 14.1 - Use a computer to investigate the family of...Ch. 14.1 - Prob. 79ECh. 14.1 - Prob. 80ECh. 14.1 - Prob. 81ECh. 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 - Prob. 12ECh. 14.2 - Prob. 13ECh. 14.2 - Prob. 14ECh. 14.2 - Prob. 15ECh. 14.2 - Prob. 16ECh. 14.2 - Prob. 17ECh. 14.2 - Prob. 18ECh. 14.2 - Prob. 19ECh. 14.2 - Prob. 20ECh. 14.2 - Prob. 21ECh. 14.2 - Prob. 22ECh. 14.2 - Prob. 23ECh. 14.2 - Prob. 24ECh. 14.2 - Prob. 25ECh. 14.2 - Find h(x,y)=g(f(x,y)) and the set of points at...Ch. 14.2 - Prob. 27ECh. 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 - Determine the set of points at which the function...Ch. 14.2 - Prob. 35ECh. 14.2 - Prob. 36ECh. 14.2 - Prob. 37ECh. 14.2 - Prob. 38ECh. 14.2 - Prob. 39ECh. 14.2 - Prob. 40ECh. 14.2 - Prob. 41ECh. 14.2 - Prob. 42ECh. 14.2 - Prob. 43ECh. 14.2 - Prob. 44ECh. 14.2 - Prob. 45ECh. 14.2 - Prob. 46ECh. 14.3 - Prob. 1ECh. 14.3 - At the beginning of this section we discussed the...Ch. 14.3 - Prob. 3ECh. 14.3 - The wave heights h in the open sea depend on the...Ch. 14.3 - Prob. 5ECh. 14.3 - Prob. 6ECh. 14.3 - Prob. 7ECh. 14.3 - Prob. 8ECh. 14.3 - Prob. 9ECh. 14.3 - A contour map is given for a function f. Use it to...Ch. 14.3 - If f(x,y)=164x2y2, find fx(1,2) and fy(1,2) and...Ch. 14.3 - If f(x,y)=4x24y2, find fx(1,0) and fy(1,0) and...Ch. 14.3 - Find fx and fy and graph f, fx, and fy with...Ch. 14.3 - Find fx and fy and graph f, fx, and fy with...Ch. 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 - Find the first partial derivatives of the...Ch. 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 - Find the first partial derivatives of the...Ch. 14.3 - Prob. 36ECh. 14.3 - Prob. 37ECh. 14.3 - Prob. 38ECh. 14.3 - Prob. 39ECh. 14.3 - Prob. 40ECh. 14.3 - Prob. 41ECh. 14.3 - Prob. 42ECh. 14.3 - Prob. 43ECh. 14.3 - Prob. 44ECh. 14.3 - Prob. 45ECh. 14.3 - Prob. 46ECh. 14.3 - Prob. 47ECh. 14.3 - Prob. 48ECh. 14.3 - Use implicit differentiation to find z/x and z/y....Ch. 14.3 - Prob. 50ECh. 14.3 - Find z/y and z/y. a z=f(x)+g(y) b z=f(x+y)Ch. 14.3 - Find z/y and z/y. a z=f(x)g(y) b z=f(xy) c...Ch. 14.3 - Prob. 53ECh. 14.3 - Prob. 54ECh. 14.3 - Prob. 55ECh. 14.3 - Prob. 56ECh. 14.3 - Prob. 57ECh. 14.3 - Prob. 58ECh. 14.3 - Prob. 59ECh. 14.3 - Prob. 60ECh. 14.3 - Verify that the conclusion of Clairauts Theorem...Ch. 14.3 - Prob. 62ECh. 14.3 - Prob. 63ECh. 14.3 - Prob. 64ECh. 14.3 - Find the indicated partial derivatives....Ch. 14.3 - Prob. 66ECh. 14.3 - Prob. 67ECh. 14.3 - Prob. 68ECh. 14.3 - Prob. 69ECh. 14.3 - Find the indicated partial derivatives. u=xaybzc;...Ch. 14.3 - Prob. 71ECh. 14.3 - If g(x,y,z)=1+xz+1xy, find gxzy. Hint: Use a...Ch. 14.3 - Use the table of values of f(x,y) to estimate the...Ch. 14.3 - Prob. 74ECh. 14.3 - Verify that the function u=e2k2tsinkx is a...Ch. 14.3 - Determine whether each of the following functions...Ch. 14.3 - Verify that the function u=1/x2+y2+z2 is a...Ch. 14.3 - Prob. 78ECh. 14.3 - If f and g are twice differentiable functions of a...Ch. 14.3 - Prob. 80ECh. 14.3 - Prob. 81ECh. 14.3 - Prob. 82ECh. 14.3 - Prob. 83ECh. 14.3 - Prob. 84ECh. 14.3 - Prob. 85ECh. 14.3 - Prob. 86ECh. 14.3 - Prob. 87ECh. 14.3 - Prob. 88ECh. 14.3 - For the ideal gas of Exercise 88, show that...Ch. 14.3 - Prob. 90ECh. 14.3 - Prob. 91ECh. 14.3 - Prob. 92ECh. 14.3 - In the project on page 271 we expressed the power...Ch. 14.3 - Prob. 94ECh. 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 - The paraboloid z=6xx22y2 intersects the plane x=1...Ch. 14.3 - The ellipsoid 4x2+2y2+z2=16 intersects the plane...Ch. 14.3 - In a study of frost penetration it was found that...Ch. 14.3 - Prob. 101ECh. 14.3 - Prob. 102ECh. 14.3 - If f(x,y)=x(x2+y2)3/2esin(x2y) find fx(1,0). Hint:...Ch. 14.3 - Prob. 104ECh. 14.3 - Prob. 105ECh. 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 - Prob. 7ECh. 14.4 - Prob. 8ECh. 14.4 - Draw the graph of f and its tangent plane at the...Ch. 14.4 - Prob. 10ECh. 14.4 - Explain why the function is differentiable at the...Ch. 14.4 - Explain why the function is differentiable at the...Ch. 14.4 - Prob. 13ECh. 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)....Ch. 14.4 - Verify the linear approximation at (0,0)....Ch. 14.4 - Given that f is a differentiable function with...Ch. 14.4 - Find the linear approximation of the function...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 - Prob. 23ECh. 14.4 - Prob. 24ECh. 14.4 - Find the differential of the function. z=e2xcos2tCh. 14.4 - Find the differential of the function. u=x2+3y2Ch. 14.4 - Find the differential of the function. m=p5q3Ch. 14.4 - Find the differential of the function. T=v1+uvwCh. 14.4 - Prob. 29ECh. 14.4 - Find the differential of the function. L=xzey2z2Ch. 14.4 - Prob. 31ECh. 14.4 - Prob. 32ECh. 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. 36ECh. 14.4 - Prob. 37ECh. 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 - In Exercise 14.1.39 and Example 14.3.3, the body...Ch. 14.4 - Prob. 42ECh. 14.4 - Show that the function is differentiable by...Ch. 14.4 - Show that the function is differentiable by...Ch. 14.4 - Prob. 45ECh. 14.4 - Prob. 46ECh. 14.5 - Use the Chain Rule to find dz/dt or dw/dt....Ch. 14.5 - Use the Chain Rule to find dz/dt or dw/dt....Ch. 14.5 - Use the Chain Rule to find dz/dt or dw/dt....Ch. 14.5 - Use the Chain Rule to find dz/dt or dw/dt....Ch. 14.5 - Use the Chain Rule to find dz/dt or dw/dt....Ch. 14.5 - Use the Chain Rule to find dz/dt or dw/dt....Ch. 14.5 - Use the Chain Rule to find z/s and z/t....Ch. 14.5 - Use the Chain Rule to find z/s and z/t....Ch. 14.5 - Use the Chain Rule to find z/s and z/t....Ch. 14.5 - Use the Chain Rule to find z/s and z/t....Ch. 14.5 - Use the Chain Rule to find z/s and z/t....Ch. 14.5 - Prob. 12ECh. 14.5 - Prob. 13ECh. 14.5 - Let R(s,t)=G(u(s,t),v(s,t)), where G, u and v are...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. 17ECh. 14.5 - Prob. 18ECh. 14.5 - Prob. 19ECh. 14.5 - Prob. 20ECh. 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 - Prob. 23ECh. 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 6 to find dy/dx. ycosx=x2+y2Ch. 14.5 - Use Equation 6 to find dy/dx. cos(xy)=1+sinyCh. 14.5 - Prob. 29ECh. 14.5 - Prob. 30ECh. 14.5 - Use Equations 7 to find z/x and z/y. x2+2y2+3z2=1Ch. 14.5 - Prob. 32ECh. 14.5 - Prob. 33ECh. 14.5 - Prob. 34ECh. 14.5 - The temperature at a point x, y is Tx, y, measured...Ch. 14.5 - Prob. 36ECh. 14.5 - The speed of sound traveling through ocean water...Ch. 14.5 - Prob. 38ECh. 14.5 - Prob. 39ECh. 14.5 - Prob. 40ECh. 14.5 - The pressure of 1 mole of an ideal gas is...Ch. 14.5 - Prob. 42ECh. 14.5 - Prob. 43ECh. 14.5 - Prob. 44ECh. 14.5 - Assume that all the given functions are...Ch. 14.5 - Prob. 46ECh. 14.5 - Prob. 47ECh. 14.5 - Prob. 48ECh. 14.5 - Prob. 49ECh. 14.5 - Prob. 50ECh. 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 - Prob. 57ECh. 14.5 - Suppose that the equation F(x,y,z)=0 implicitly...Ch. 14.5 - Prob. 59ECh. 14.6 - Level curves for barometric pressure in millibars...Ch. 14.6 - The contour map shows the average maximum...Ch. 14.6 - A table of values for the wind-chill index...Ch. 14.6 - Find the directional derivative of f at the given...Ch. 14.6 - Prob. 5ECh. 14.6 - Find the directional derivative of f at the given...Ch. 14.6 - a Find the gradient of f. b Evaluate the gradient...Ch. 14.6 - Prob. 8ECh. 14.6 - Prob. 9ECh. 14.6 - Prob. 10ECh. 14.6 - Prob. 11ECh. 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. 15ECh. 14.6 - Prob. 16ECh. 14.6 - Find the directional derivative of the function at...Ch. 14.6 - Use the figure to estimate Duf(2,2).Ch. 14.6 - Prob. 19ECh. 14.6 - Prob. 20ECh. 14.6 - Prob. 21ECh. 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 - Prob. 24ECh. 14.6 - Prob. 25ECh. 14.6 - Find the maximum rate of change of f at the given...Ch. 14.6 - a Show that a differentiable function f decreases...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. 32ECh. 14.6 - Prob. 33ECh. 14.6 - Suppose you are climbing a hill whose shape is...Ch. 14.6 - Prob. 35ECh. 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 function...Ch. 14.6 - The second directional derivative of f(x,y) is...Ch. 14.6 - a If u=a,b is a unit vector and f has continuous...Ch. 14.6 - Find equations of a the tangent plane and b the...Ch. 14.6 - Prob. 42ECh. 14.6 - Prob. 43ECh. 14.6 - Prob. 44ECh. 14.6 - Prob. 45ECh. 14.6 - Prob. 46ECh. 14.6 - Prob. 47ECh. 14.6 - Prob. 48ECh. 14.6 - If f(x,y)=xy, find the gradient vector f(3,2) and...Ch. 14.6 - Prob. 50ECh. 14.6 - Prob. 51ECh. 14.6 - Prob. 52ECh. 14.6 - Prob. 53ECh. 14.6 - At what point on the ellipsoid x2+y2+2z2=1 is the...Ch. 14.6 - Are there any points on the hyperboloid x2y2z2=1...Ch. 14.6 - Show that the ellipsoid 3x2+2y2+z2=9 and the...Ch. 14.6 - Prob. 57ECh. 14.6 - Show that every normal line to the sphere...Ch. 14.6 - Prob. 59ECh. 14.6 - Prob. 60ECh. 14.6 - Show that the sum of the x-, y-, and z-intercepts...Ch. 14.6 - Prob. 62ECh. 14.6 - Find parametric equations for the tangent line to...Ch. 14.6 - a The plane y+z=3 intersects the cylinder x2+y2=5...Ch. 14.6 - Where does the helix r(t)=cost,sint,t intersect...Ch. 14.6 - Prob. 66ECh. 14.6 - Prob. 67ECh. 14.6 - Prob. 68ECh. 14.6 - Suppose that the directional derivatives of f(x,y)...Ch. 14.6 - Prob. 70ECh. 14.7 - Prob. 1ECh. 14.7 - Suppose 0, 2 is a critical point of a function g...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 - Prob. 6ECh. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Prob. 8ECh. 14.7 - Prob. 9ECh. 14.7 - Prob. 10ECh. 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 - Prob. 16ECh. 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. 19ECh. 14.7 - Prob. 20ECh. 14.7 - Prob. 21ECh. 14.7 - Show that f(x,y)=x2yex2y2 has minimum values at...Ch. 14.7 - Prob. 23ECh. 14.7 - Prob. 24ECh. 14.7 - Use a graph or level curves or both to estimate...Ch. 14.7 - Prob. 26ECh. 14.7 - Prob. 27ECh. 14.7 - Use a graphing device as in Example 4 or Newtons...Ch. 14.7 - Prob. 29ECh. 14.7 - Prob. 30ECh. 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 - Prob. 34ECh. 14.7 - Prob. 35ECh. 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 - Prob. 38ECh. 14.7 - Prob. 39ECh. 14.7 - Prob. 40ECh. 14.7 - Find the shortest distance from the point (2,0,3)...Ch. 14.7 - Prob. 42ECh. 14.7 - Prob. 43ECh. 14.7 - Prob. 44ECh. 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 - Prob. 49ECh. 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 - Prob. 56ECh. 14.7 - The Shannon index sometimes called the...Ch. 14.7 - Prob. 58ECh. 14.7 - Suppose that a scientist has reason to believe...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 - Prob. 2ECh. 14.8 - Each of these extreme value problems has a...Ch. 14.8 - Prob. 4ECh. 14.8 - Prob. 5ECh. 14.8 - Prob. 6ECh. 14.8 - Prob. 7ECh. 14.8 - Prob. 8ECh. 14.8 - Each of these extreme value problems has a...Ch. 14.8 - Prob. 10ECh. 14.8 - Each of these extreme value problems has a...Ch. 14.8 - Prob. 12ECh. 14.8 - Prob. 13ECh. 14.8 - Each of these extreme value problems has a...Ch. 14.8 - The method of Lagrange multipliers assumes that...Ch. 14.8 - Find the minimum value of f(x,y,z)=x2+2y2+3z2...Ch. 14.8 - Prob. 17ECh. 14.8 - Prob. 18ECh. 14.8 - Prob. 19ECh. 14.8 - Prob. 20ECh. 14.8 - Find the extreme values of f on the region...Ch. 14.8 - Find the extreme values of f on the region...Ch. 14.8 - Find the extreme values of f on the region...Ch. 14.8 - Prob. 24ECh. 14.8 - Consider the problem of minimizing the function...Ch. 14.8 - Prob. 26ECh. 14.8 - The total production P of a certain product...Ch. 14.8 - Prob. 28ECh. 14.8 - Use Lagrange multipliers to prove 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 - Prob. 32ECh. 14.8 - Prob. 33ECh. 14.8 - Prob. 34ECh. 14.8 - Use Lagrange multipliers to give an alternate...Ch. 14.8 - Prob. 36ECh. 14.8 - Prob. 37ECh. 14.8 - Prob. 38ECh. 14.8 - Prob. 39ECh. 14.8 - Prob. 40ECh. 14.8 - Prob. 41ECh. 14.8 - Prob. 42ECh. 14.8 - Prob. 43ECh. 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 4x3y+8z=5 intersects the cone z2=x2+y2...Ch. 14.8 - Prob. 47ECh. 14.8 - Find the maximum and minimum values of f subject...Ch. 14.8 - a Find the maximum value of...Ch. 14.8 - a Maximize i=1nxiyi subject to the constraints...Ch. 14.R - Prob. 1CCCh. 14.R - Prob. 2CCCh. 14.R - Prob. 3CCCh. 14.R - Prob. 4CCCh. 14.R - Prob. 5CCCh. 14.R - What does Clairauts Theorem say?Ch. 14.R - Prob. 7CCCh. 14.R - Prob. 8CCCh. 14.R - Prob. 9CCCh. 14.R - Prob. 10CCCh. 14.R - Prob. 11CCCh. 14.R - Prob. 12CCCh. 14.R - Prob. 13CCCh. 14.R - aDefine the gradient vector f for a function f of...Ch. 14.R - Prob. 15CCCh. 14.R - Prob. 16CCCh. 14.R - Prob. 17CCCh. 14.R - Prob. 18CCCh. 14.R - Prob. 19CCCh. 14.R - Prob. 1TFQCh. 14.R - Prob. 2TFQCh. 14.R - Prob. 3TFQCh. 14.R - Prob. 4TFQCh. 14.R - Prob. 5TFQCh. 14.R - Prob. 6TFQCh. 14.R - Prob. 7TFQCh. 14.R - Prob. 8TFQCh. 14.R - Prob. 9TFQCh. 14.R - Prob. 10TFQCh. 14.R - Prob. 11TFQCh. 14.R - Prob. 12TFQCh. 14.R - Prob. 1ECh. 14.R - Prob. 2ECh. 14.R - Prob. 3ECh. 14.R - Prob. 4ECh. 14.R - Prob. 5ECh. 14.R - Prob. 6ECh. 14.R - Make a rough sketch of a contour map for the...Ch. 14.R - The contour map of a function f is shown. a...Ch. 14.R - Prob. 9ECh. 14.R - Prob. 10ECh. 14.R - Prob. 11ECh. 14.R - Prob. 12ECh. 14.R - Prob. 13ECh. 14.R - Prob. 14ECh. 14.R - Prob. 15ECh. 14.R - Prob. 16ECh. 14.R - Prob. 17ECh. 14.R - Prob. 18ECh. 14.R - Prob. 19ECh. 14.R - Prob. 20ECh. 14.R - Prob. 21ECh. 14.R - Prob. 22ECh. 14.R - Prob. 23ECh. 14.R - Prob. 24ECh. 14.R - Prob. 25ECh. 14.R - Prob. 26ECh. 14.R - Prob. 27ECh. 14.R - Find equations of a the tangent plane and b the...Ch. 14.R - Find equations of a the tangent plane and b the...Ch. 14.R - Prob. 30ECh. 14.R - Find the points on the hyperboloid x2+4y2z2=4...Ch. 14.R - Prob. 32ECh. 14.R - Prob. 33ECh. 14.R - The two legs of a right triangle are measured as 5...Ch. 14.R - Prob. 35ECh. 14.R - Prob. 36ECh. 14.R - Suppose z=f(x,y), where x=g(s,t), y=h(s,t),...Ch. 14.R - Prob. 38ECh. 14.R - Prob. 39ECh. 14.R - Prob. 40ECh. 14.R - Prob. 41ECh. 14.R - Prob. 42ECh. 14.R - Prob. 43ECh. 14.R - Prob. 44ECh. 14.R - Prob. 45ECh. 14.R - Prob. 46ECh. 14.R - Prob. 47ECh. 14.R - Prob. 48ECh. 14.R - The contour map shows wind speed in knots during...Ch. 14.R - Prob. 50ECh. 14.R - Prob. 51ECh. 14.R - Prob. 52ECh. 14.R - Prob. 53ECh. 14.R - Prob. 54ECh. 14.R - Prob. 55ECh. 14.R - Prob. 56ECh. 14.R - Use a graph or level curves or both to estimate...Ch. 14.R - Use a graphing calculator or computer or Newtons...Ch. 14.R - Prob. 59ECh. 14.R - Prob. 60ECh. 14.R - Prob. 61ECh. 14.R - Prob. 62ECh. 14.R - Prob. 63ECh. 14.R - A package in the shape of a rectangular box can be...Ch. 14.R - A pentagon is formed by placing an isosceles...Ch. 14.P - A rectangle with length L and width W is cut into...Ch. 14.P - Marine biologists have determined that when a...Ch. 14.P - Prob. 3PCh. 14.P - For what values of the number r is the function...Ch. 14.P - Prob. 5PCh. 14.P - Prob. 6PCh. 14.P - If the ellipse x2/a2+y2/b2=1 is to enclose the...Ch. 14.P - Show that the maximum value of the function...
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