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mathematics
applied calculus

Calculus And Its Applications 14th Edition Larry Goldstein, David Lay, David Schneider, Nakhle Asmar - Solutions

Use logarithmic differentiation to differentiate the following functions.f (x) = 2x
Solve for t.e0.05t - 4e-0.06t = 0
Solve the following equations for t.3e2t = 15
Sketch the graphs of the following functions.y = 1 - ex
Use logarithmic differentiation to differentiate the following functions.f (x) = x√3
Solve for t.4e0.01t - 3e0.04t = 0
Solve the following equations for t.3et/2 - 12 = 0
Sketch the graphs of the following functions.y = ex/2
Use logarithmic differentiation to differentiate the following functions.f (x) = xx
Solve the following equations for t.2 ln t = 5
Sketch the graphs of the following functions.y = ex-1
Use logarithmic differentiation to differentiate the following functions.f (x) = x√x
Under certain geographic conditions, the wind velocity v at a height x centimeters above the ground is given by v = K ln(x/x0), where K is a positive constant (depending on the air density, average wind velocity, and the like) and x0 is a roughness parameter (depending on the roughness of the
Solve the following equations for t.2e- 0.3t = 1
Sketch the graphs of the following functions.y = -e-x + 1
Substantial empirical data show that, if x and y measure the sizes of two organs of a particular animal, then x and y are related by an allometric equation of the form ln y - k ln x = ln c, where k and c are positive constants that depend only on the type of parts or organs that are measured and
Find k such that 2x = ekx for all x.
Sketch the graphs of the following functions.y = 2e-x
In the study of epidemics, we find the equation ln(1 - y) - ln y = C - rt, where y is the fraction of the population that has a specific disease at time t. Solve the equation for y in terms of t and the constants C and r.
Find k such that 2-x/5 = ekx for all x.
Use logarithmic differentiation to differentiate the following functions. st x+5x + 1 I x [ + st 5 (x)ƒ
Calculate values offor small values of x, and use them to estimateWhat is the formula for 10% - 1 X
Graph y = ln 5x and y = 2 together and determine the x-coordinate of their point of intersection (to four decimal places). Express this number in terms of a power of e.
Set Y1 = ex and use your calculator’s derivative command to specify Y2 as the derivative of Y1. Graph the two functions simultaneously in the window [-1, 3] by [-3, 20] and observe that the graphs overlap.
Graph y = e2x and y = 5 together, and determine the xcoordinate of their point of intersection (to four decimal places). Express this number in terms of a logarithm.
(a) Graph y = ex.(b) Zoom in on the region near x = 0 until the curve appears as a straight line and estimate the slope of the line. This number is an estimate ofCompare your answer with the actual slope, 1.(c) Repeat parts (a) and (b) for y = 2x. Observe that the slope at x = 0 is not 1. d dx -et
Find values of k and r for which the graph of y = kxr passes through the points (2, 3) and (4, 15).
Graph the function y = ln(ex), and use trace to convince yourself that it is the same as the function y = x. What do you observe about the graph of y = eln x?
Determine the values of h and k for which the graph of y = hekx passes through the points (1, 6) and (4, 48).
Find the equation of the tangent line to the graph of y = ex at x = 0. Then, graph the function and the tangent line together to confirm that your answer is correct.
Use logarithmic differentiation to differentiate the following functions. f(x) = (x² + 5)6(x³ + 7)8(x4 + 9)¹0
Use logarithmic differentiation to differentiate the following functions. f(x) = bx, where b > 0
Use logarithmic differentiation to differentiate the following functions. f(x) = xVx
Use logarithmic differentiation to differentiate the following functions. f(x) = 2x
Use logarithmic differentiation to differentiate the following functions. 1+x x¹+ f(x) = x
Use logarithmic differentiation to differentiate the following functions. f(x) = 10x
Use logarithmic differentiation to differentiate the following functions. 2 f(x) = √x² + 5 et²
Use logarithmic differentiation to differentiate the following functions. f(x) = Xex V₁-3 x³ + 3
Use logarithmic differentiation to differentiate the following functions. f(x) = ·x√x + 1(x² + 2x + 3)² 4x²
Use logarithmic differentiation to differentiate the following functions. f(x) = ex+¹(x² + 1)x
Use logarithmic differentiation to differentiate the following functions. f(x) = exx²2x
The atmospheric pressure at an altitude of x kilometers is f (x) g/cm2 (grams per square centimeter), where f(x) = 1035e-0.12x. Give approximate answers to the following questions using the graphs of f (x) and f′(x) shown in Fig. 2.(a) What is the pressure at an altitude of 2 kilometers?(b) At
The health expenditures (in billions of dollars) for a certain country from 1990 to 2010 are given approximately by f (t) = 27e0.106t, with time in years measured from 1990. Give approximate answers to the following questions using the graphs of f (t) and f′(t) shown in Fig. 3.(a) How much money
Outline a method for locating the relative extreme points of a function.
Sketch the graphs of the following functions.f(x) = -x3
If f (25) = 10 and f′(25) = -2, estimate each of the following.(a) f(27)(b) f(26)(c) f(25.25) (d) f(24)(e) f(23.5)
Sketch the graph of a function that has the properties described.(-2, -1) and (2, 5) are on the graph; f′(-2) = 0 and f′(2) = 0; f′(x) > 0 for x < 0, f′(0) = 0, f′(x) < 0 for x > 0.
Use the approach of Exercise 77 to show thatfor any constant c.Exercise 77Draw two graphs of your choice that represent a function y = f (x) and its vertical shift y = f (x) + 3.Pick a value of x and consider the points (x, f (x)) and (x, f (x) + 3). Draw the tangent lines to the curves at these
Figure 2 shows the graph of the function f(x) and its tangent line at x = 3. Find f (3), f′(3), and f″(3). 5 4 3 2 1 Figure 2 y = f(x) 2 3 4 5 תיו x
Each of the graphs of the functions in Exercises has one relative extreme point. Plot this point and check the concavity there. Using only this information, sketch the graph. [Recall that if f (x) = ax2 + bx + c, then f(x) has a relative minimum point when a > 0 and a relative maximum point when
Refer to the inventory problem of Example 2. If the distributor offers a discount of $1 per case for orders of 600 or more cases, should the manager change the quantity ordered?Example 2Suppose that the manager in Example 1 wants to establish an optimal inventory policy for frozen orange juice.
Find the positive values of x, y, and z that maximize Q = xyz, if x + y = 1 and y + z = 2. What is this maximum value?
The relationship between the area of the pupil of the eye and the intensity of light was analyzed by B. H. Crawford. Crawford concluded that the area of the pupil issquare millimeters when x units of light are entering the eye per unit time.(a) Graph f(x) and f′(x) in the window [0, 6] by [-5,
In Exercises, find dy/dx, where y is a function of u such thatState the answer in terms of x only.u = x3/2 dy U du +1 u²
Exercises refer to the graph in Fig. 3. List the labeled values of x at which the derivative has the stated property.f″(x) is negative. y Figure 3 a b I c d e y = f(x) X
Describe each of the following graphs. Your descriptions should include each of the six categories mentioned previously.
Exercises refer to the graph in Fig. 3. List the labeled values of x at which the derivative has the stated property.f′(x) is maximized. y Figure 3 a b I c d e y = f(x) X
Each of the graphs of the functions in Exercises has one relative extreme point. Plot this point and check the concavity there. Using only this information, sketch the graph. [Recall that if f (x) = ax2 + bx + c, then f(x) has a relative minimum point when a > 0 and a relative maximum point when
Starting with a 100-foot-long stone wall, a farmer would like to construct a rectangular enclosure by adding 400 feet of fencing, as shown in Fig. 7(a). Find the values of x and w that result in the greatest possible area. 100 feet x W (a) Figure 7 Rectangular enclosures. x (b) W
Consider a rectangle in the xy-plane, with corners at (0, 0), (a, 0), (0, b), and (a, b). If (a, b) lies on the graph of the equation y = 30 - x, find a and b such that the area of the rectangle is maximized. What economic interpretations can be given to your answer if the equation y = 30 - x
There are $320 available to fence in a rectangular garden. The fencing for the side of the garden facing the road costs $6 per foot and the fencing for the other three sides costs $2 per foot. [See Fig. 12(a).] Consider the problem of finding the dimensions of the largest possible garden.(a)
Sketch the graph of a function that has the properties described.(0, 6), (2, 3), and (4, 0) are on the graph; f′(0) = 0 and f′(4) = 0; f′(x) < 0 for x < 2, f′(2) = 0, f′(x) > 0 for x > 2.
Sketch the graphs of the following functions.f(x) = x3 + 3x + 1
Describe each of the following graphs. Your descriptions should include each of the six categories mentioned previously.
Outline a method for locating the inflection points of a function.
Rework Exercise 11 for the case where only 200 feet of fencing is added to the stone wall.Exercise 11Starting with a 100-foot-long stone wall, a farmer would like to construct a rectangular enclosure by adding 400 feet of fencing, as shown in Fig. 7(a). Find the values of x and w that result in the
Figure 12(b) shows an open rectangular box with a square base. Consider the problem of finding the values of x and h for which the volume is 32 cubic feet and the total surface area of the box is minimal. (The surface area is the sum of the areas of the five faces of the box.)(a) Determine the
Describe each of the following graphs. Your descriptions should include each of the six categories mentioned previously. fi NA 1 X
Until recently hamburgers at the city sports arena cost $4 each. The food concessionaire sold an average of 10,000 hamburgers on a game night. When the price was raised to $4.40, hamburger sales dropped off to an average of 8000 per night.(a) Assuming a linear demand curve, find the price of a
A rectangular corral of 54 square meters is to be fenced off and then divided by a fence into two sections, as shown in Fig. 7(b). Find the dimensions of the corral so that the amount of fencing required is minimized. x (b) W
Sketch the graph of a function that has the properties described.f(x) defined only for x ≥ 0; (0, 0) and (5, 6) are on the graph; f′(x) > 0 for x ≥ 0; f′ (x) < 0 for x < 5, f′′(5) = 0, f′′(x) > 0 for x > 5.
Exercises refer to the graph in Fig. 3. List the labeled values of x at which the derivative has the stated property.f′(x) is minimized. y Figure 3 a b I c d e y = f(x) X
Each of the graphs of the functions in Exercises has one relative extreme point. Plot this point and check the concavity there. Using only this information, sketch the graph. [Recall that if f (x) = ax2 + bx + c, then f(x) has a relative minimum point when a > 0 and a relative maximum point when
Sketch the graphs of the following functions.f(x) = x3 + 2x2 + 4x
Refer to Exercise 13. If the cost of the fencing for the boundary is $5 per meter and the dividing fence costs $2 per meter, find the dimensions of the corral that minimize the cost of the fencing.Exercise 13A rectangular corral of 54 square meters is to be fenced off and then divided by a fence
Outline a procedure for sketching the graph of a function.
Describe the way the slope changes on the graph in Exercise 6.Describe each of the following graphs. Your descriptions should include each of the six categories mentioned previously. y 10 H X
Each of the graphs of the functions in Exercises has one relative extreme point. Plot this point and check the concavity there. Using only this information, sketch the graph. [Recall that if f (x) = ax2 + bx + c, then f(x) has a relative minimum point when a > 0 and a relative maximum point when
Properties of various functions are described next. In each case, draw some conclusion about the graph of the function. g(1) = 5, g'(1) = -1
The average ticket price for a concert at the opera house was $50. The average attendance was 4000. When the ticket price was raised to $52, attendance declined to an average of 3800 persons per performance. What should the ticket price be to maximize the revenue for the opera house? (Assume a
A large soup can is to be designed so that the can will hold 16π cubic inches (about 28 ounces) of soup. [See Fig. 14(b).] Find the values of x and h for which the amount of metal needed is as small as possible. 2TX Side unrolled (b) h
Sketch the graphs of the following functions. f(x)= x³2x² + x
Developers from two cities, A and B, want to connect their cities to a major highway and plan to build rest stops and gas stations at the highway entrance. To minimize the cost of road construction, the developers must find the location for the highway entrance that minimizes the total distance, d1
Simultaneously graph the functionsin the window [-6, 6] by [-6, 6]. Describe the asymptote of the first function. 1 y = =+ x and y = x X
Consider the problem of finding the dimensions of the rectangular garden of area 100 square meters for which the amount of fencing needed to surround the garden is as small as possible.(a) Draw a picture of a rectangle and select appropriate letters for the dimensions.(b) Determine the objective
Figure 14(a) shows a Norman window, which consists of a rectangle capped by a semicircular region. Find the value of x such that the perimeter of the window will be 14 feet and the area of the window will be as large as possible. x 2x (a) Figure 14 h 1 X h (b) 2п х Side unrolled h
A swimming club offers memberships at the rate of $200, provided that a minimum of 100 people join. For each member in excess of 100, the membership fee will be reduced $1 per person (for each member). At most, 160 memberships will be sold. How many memberships should the club try to sell to
A certain toll road averages 36,000 cars per day when charging $1 per car. A survey concludes that increasing the toll will result in 300 fewer cars for each cent of increase. What toll should be charged to maximize the revenue?
Differentiate the functions in Exercises. y=(x+1)(x3+5x+2)
Differentiate the functions in Exercises. y = (-x^ + 2)| +2)( 2 1
Match each observation (a)–(e) with a conclusion (A)–(E).Observations(a) The point (3, 4) is on the graph of f′(x).(b) The point (3, 4) is on the graph of f (x).(c) The point (3, 4) is on the graph of f′(x).(d) The point (3, 0) is on the graph of f′(x), and the point (3, 4) is on the
Compute f(g(x)), where f(x) and g(x) are the following: f(x) = X T² 8(x) = x³ x + 1'
Compute f(g(x)), where f(x) and g(x) are the following: f(x)=x-1, g(x) = 1 x + 1
Suppose that the cost function in Exercise 45 is C(x) = -2.5x + 1, where x% is the index-fund fee. (The company has a fixed cost of $1 billion, and the cost decreases as a function of the index-fund fee.) Find the value of x that maximizes profit. How well did Fidelity Mutual do before and after
Let f (x) be the number of people living within x miles of the center of New York City.(a) What does f(10 + h) - f (10) represent?(b) Explain why f′(10) cannot be negative.
State the product rule and the quotient rule.
Compute f(g(x)), where f(x) and g(x) are the following: f(x) = x(x² + 1), g(x) = √x
In Exercises, suppose that x and y are related by the given equation and use implicit differentiation to determine dy/dx.x2 - y2 = 1
Differentiate the functions in Exercises. y = (2x² = x + 1)(-x³ + 1) -
Differentiate the functions in Exercises. y = (x² + x + 1)²(x - 1)4

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