Calculus & Its Applications (14th Edition)
Calculus & Its Applications (14th Edition)
14th Edition
ISBN: 9780134437774
Author: Larry J. Goldstein, David C. Lay, David I. Schneider, Nakhle H. Asmar
Publisher: PEARSON
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Unitary Method
The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products.
Speed, Time, and Distance
Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 …
Profit and Loss
The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.
Units and Measurements
Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experime…
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Chapter 4.3, Problem 33E

An Investment Portfolio The value of an investment portfolio consisting of two stocks is given by f ( t ) = 3 e 0.06 t + 2 e 0.02 t , where t is the number of years since the inception of the portfolio, and f ( t ) is in thousands of dollars.

a. What is the initial dollar amount invested?

b. What is the value of the portfolio after 5 years ?

c. At what rate is the investment appreciating after 5 years ?

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Chapter 4 Solutions

Calculus & Its Applications (14th Edition)

Ch. 4.1 - Can a function such as f(x)=53x be written in the...Ch. 4.1 - Solve the equation 7263x=28.Ch. 4.1 - Prob. 1ECh. 4.1 - Prob. 2ECh. 4.1 - Write each expression in Exercises 1-14 in the...Ch. 4.1 - Write each expression in Exercises 1-14 in the...Ch. 4.1 - Write each expression in Exercises 1-14 in the...Ch. 4.1 - Write each expression in Exercises 1-14 in the...Ch. 4.1 - Prob. 7ECh. 4.1 - Write each expression in Exercises 1-14 in the...
Ch. 4.1 - Write each expression in Exercises 1-14 in the...Ch. 4.1 - Prob. 10ECh. 4.1 - Prob. 11ECh. 4.1 - Write each expression in Exercises 1-14 in the...Ch. 4.1 - Prob. 13ECh. 4.1 - Prob. 14ECh. 4.1 - Find a number b such that the function f(x)=32x...Ch. 4.1 - Find b so that 8x/3=bx for all x.Ch. 4.1 - Solve the following equations for x. 52x=52Ch. 4.1 - Solve the following equations for x. 10x=102Ch. 4.1 - Solve the following equations for x....Ch. 4.1 - Solve the following equations for x....Ch. 4.1 - Solve the following equations for x. 101x=100Ch. 4.1 - Solve the following equations for x. 24x=8Ch. 4.1 - Solve the following equations for x. 3(2.7)5x=8.1Ch. 4.1 - Solve the following equations for x....Ch. 4.1 - Solve the following equations for x. (2x+123)2=2Ch. 4.1 - Solve the following equations for x. (32x32)4=3Ch. 4.1 - Solve the following equations for x. 23x=425xCh. 4.1 - Solve the following equations for x. 35x3x3=0Ch. 4.1 - Solve the following equations for x. (1+x)2x52x=0Ch. 4.1 - Prob. 30ECh. 4.1 - Solve the following equations for x. 2x822x=0Ch. 4.1 - Prob. 32ECh. 4.1 - Solve the following equations for x. [Hint: In...Ch. 4.1 - Prob. 34ECh. 4.1 - Solve the following equations for x. [Hint: In...Ch. 4.1 - Prob. 36ECh. 4.1 - The expressions in Exercises 37-42 may be factored...Ch. 4.1 - The expressions in Exercises 37-42 may be factored...Ch. 4.1 - The expressions in Exercises 37-42 may be factored...Ch. 4.1 - The expressions in Exercises 37-42 may be factored...Ch. 4.1 - The expressions in Exercises 37-42 may be factored...Ch. 4.1 - Prob. 42ECh. 4.1 - Prob. 43ECh. 4.1 - Prob. 44ECh. 4.1 - Prob. 45ECh. 4.2 - Solve the following equation for x: e6x=e3.Ch. 4.2 - Differentiate y=(x+ex)4Ch. 4.2 - Show that ddx(3x)|x=01.1 by calculating the slope...Ch. 4.2 - Show that ddx(2.7x)|x=0.99 by calculating the...Ch. 4.2 - In Exercises 3-6, compute the given derivatives...Ch. 4.2 - Prob. 4ECh. 4.2 - Prob. 5ECh. 4.2 - Prob. 6ECh. 4.2 - Write each expression in the form ekx for a...Ch. 4.2 - Write each expression in the form ekx for a...Ch. 4.2 - Write each expression in the form ekx for a...Ch. 4.2 - Write each expression in the form ekx for a...Ch. 4.2 - Write each expression in the form ekx for a...Ch. 4.2 - Prob. 12ECh. 4.2 - Solve each equation for x. e5x=e20Ch. 4.2 - Prob. 14ECh. 4.2 - Solve each equation for x. ex22x=e8Ch. 4.2 - Prob. 16ECh. 4.2 - Solve each equation for x. ex(x21)=0Ch. 4.2 - Solve each equation for x. 4ex(x2+1)=0Ch. 4.2 - Find an equation of the tangent line to the graph...Ch. 4.2 - Prob. 20ECh. 4.2 - Use the first and second derivative rules from...Ch. 4.2 - Prob. 22ECh. 4.2 - Suppose that A=(a,b) is a point on the graph of...Ch. 4.2 - Find the slope-point form of the equation of the...Ch. 4.2 - Differentiate the following functions. y=3ex7xCh. 4.2 - Differentiate the following functions. y=2x+45ex4Ch. 4.2 - Differentiate the following functions. y=xexCh. 4.2 - Differentiate the following functions....Ch. 4.2 - Differentiate the following functions....Ch. 4.2 - Differentiate the following functions....Ch. 4.2 - Differentiate the following functions. y=exx+1Ch. 4.2 - Prob. 32ECh. 4.2 - Differentiate the following functions. y=ex1ex+1Ch. 4.2 - Differentiate the following functions. y=ex+1Ch. 4.2 - The graph of y=xex has one extreme point. Find its...Ch. 4.2 - Prob. 36ECh. 4.2 - Find the point on the graph of y=(1+x2)ex where...Ch. 4.2 - Prob. 38ECh. 4.2 - Find the slope of the tangent line to the curve...Ch. 4.2 - Find the slope of the tangent line to the curve...Ch. 4.2 - Find the equation of the tangent line to the curve...Ch. 4.2 - Find the equation of the tangent line to the curve...Ch. 4.2 - Find the first and second derivatives....Ch. 4.2 - Find the first and second derivatives. f(x)=exxCh. 4.2 - Compute the following derivatives. ddx(5ex)...Ch. 4.2 - Prob. 46ECh. 4.2 - Prob. 47ECh. 4.2 - Prob. 48ECh. 4.2 - Prob. 49ECh. 4.2 - Prob. 50ECh. 4.2 - Prob. 51ECh. 4.2 - Prob. 52ECh. 4.2 - Prob. 53ECh. 4.2 - Prob. 54ECh. 4.2 - Prob. 55ECh. 4.2 - Prob. 56ECh. 4.3 - Differentiate tet2Ch. 4.3 - Differentiate [ e3x(1+e6x) ]12.Ch. 4.3 - Differentiate the following functions. f(x)=e2x+3Ch. 4.3 - Differentiate the following functions. f(x)=e3x2Ch. 4.3 - Differentiate the following functions. f(x)=e4x2xCh. 4.3 - Differentiate the following functions....Ch. 4.3 - Differentiate the following functions. f(x)=eexCh. 4.3 - Differentiate the following functions. f(x)=e1xCh. 4.3 - Differentiate the following functions. f(x)=exCh. 4.3 - Differentiate the following functions. f(x)=ex2+1Ch. 4.3 - Differentiate the following functions. f(x)=7ex7Ch. 4.3 - Differentiate the following functions. f(x)=10ex25Ch. 4.3 - Differentiate the following functions....Ch. 4.3 - Differentiate the following functions....Ch. 4.3 - Differentiate the following functions....Ch. 4.3 - Differentiate the following functions....Ch. 4.3 - Differentiate the following functions....Ch. 4.3 - Differentiate the following functions. f(x)=eeexCh. 4.3 - Differentiate the following functions....Ch. 4.3 - Differentiate the following functions....Ch. 4.3 - Differentiate the following functions. f(x)=ex+1Ch. 4.3 - Differentiate the following functions. f(x)=eexCh. 4.3 - In Exercises 21-26, simplify the function before...Ch. 4.3 - In Exercises 21-26, simplify the function before...Ch. 4.3 - In Exercises 21-26, simplify the function before...Ch. 4.3 - In Exercises 21-26, simplify the function before...Ch. 4.3 - In Exercises 21-26, simplify the function before...Ch. 4.3 - In Exercises 21-26, simplify the function before...Ch. 4.3 - In Exercises 27-32, find the values of x at which...Ch. 4.3 - In Exercises 27-32, find the values of x at which...Ch. 4.3 - In Exercises 27-32, find the values of x at which...Ch. 4.3 - In Exercises 27-32, find the values of x at which...Ch. 4.3 - In Exercises 27-32, find the values of x at which...Ch. 4.3 - In Exercises 27-32, find the values of x at which...Ch. 4.3 - An Investment Portfolio The value of an investment...Ch. 4.3 - Depreciation of Assets The value of the computer t...Ch. 4.3 - The Most Expensive Artwork to Date The highest...Ch. 4.3 - Appreciation of Assets A painting purchased in...Ch. 4.3 - Velocity and Acceleration The velocity of the...Ch. 4.3 - Velocity and Acceleration Suppose the velocity of...Ch. 4.3 - Heights of a Plant The height of a certain plant,...Ch. 4.3 - Heights of a Plant The length of a certain weed,...Ch. 4.3 - Gompertz Growth Curve Let aandb be positive...Ch. 4.3 - Find dydx if y=e(110)ex2.Ch. 4.3 - Size of Tumor In a study, a cancerous tumor was...Ch. 4.3 - Height of a Plant Let f(t) be the function from...Ch. 4.4 - Find lne.Ch. 4.4 - Solve e3x=2 using the natural logarithm function.Ch. 4.4 - Find ln(e).Ch. 4.4 - Find ln(1e2).Ch. 4.4 - If ex=5, Write x in terms of the natural...Ch. 4.4 - If ex=3.2, Write x in terms of the natural...Ch. 4.4 - If lnx=1, Write x using the exponential function.Ch. 4.4 - If lnx=4.5, Write x using the exponential...Ch. 4.4 - Simplify the following expression. lne3Ch. 4.4 - Simplify the following expression. eln4.1Ch. 4.4 - Simplify the following expression. eeln1Ch. 4.4 - Simplify the following expression. ln(e2lne)Ch. 4.4 - Simplify the following expression. ln(lne)Ch. 4.4 - Simplify the following expression. e4ln1Ch. 4.4 - Simplify the following expression. e2lnxCh. 4.4 - Simplify the following expression. exln2Ch. 4.4 - Simplify the following expression. e2ln7Ch. 4.4 - Simplify the following expression. e2ln7Ch. 4.4 - Simplify the following expression. elnx+ln2Ch. 4.4 - Simplify the following expression. eln32lnxCh. 4.4 - Solve the following equations for x. e2x=5Ch. 4.4 - Solve the following equations for x. e13x=4Ch. 4.4 - Solve the following equations for x. ln(4x)=12Ch. 4.4 - Prob. 22ECh. 4.4 - Solve the following equations for x. lnx2=9Ch. 4.4 - Prob. 24ECh. 4.4 - Solve the following equations for x. 6e0.00012x=3Ch. 4.4 - Prob. 26ECh. 4.4 - Solve the following equations for x. ln3x=ln5Ch. 4.4 - Prob. 28ECh. 4.4 - Solve the following equations for x. ln(ln3x)=0Ch. 4.4 - Prob. 30ECh. 4.4 - Solve the following equations for x. 2ex/39=0Ch. 4.4 - Prob. 32ECh. 4.4 - Prob. 33ECh. 4.4 - Prob. 34ECh. 4.4 - Prob. 35ECh. 4.4 - Prob. 36ECh. 4.4 - Solve the following equations for x. 4exe2x=6Ch. 4.4 - Prob. 38ECh. 4.4 - The graph of f(x)=5x+ex is shown in fig. 4. Find...Ch. 4.4 - Prob. 40ECh. 4.4 - Prob. 41ECh. 4.4 - Prob. 42ECh. 4.4 - Prob. 43ECh. 4.4 - Find the x-intercept of y=(x1)2ln(x+1),x1.Ch. 4.4 - In Exercise 45- 46, find the coordinates of each...Ch. 4.4 - In Exercise 45- 46, find the coordinates of each...Ch. 4.4 - Solve for t. e0.05t4e0.06t=0Ch. 4.4 - Solve for t. 4e0.01t3e0.04t=0Ch. 4.4 - Prob. 49ECh. 4.4 - Wind Velocity Under certain geographic conditions,...Ch. 4.4 - Prob. 51ECh. 4.4 - Prob. 52ECh. 4.4 - Prob. 53ECh. 4.4 - Prob. 54ECh. 4.4 - Prob. 55ECh. 4.5 - Differentiate f(x)=1ln(x4+5).Ch. 4.5 - Differentiate f(x)=ln(lnx).Ch. 4.5 - Differentiate the following functions. y=3lnx+ln2Ch. 4.5 - Differentiate the following functions. y=lnxln3Ch. 4.5 - Differentiate the following functions. y=x2lnx2Ch. 4.5 - Differentiate the following functions. y=3lnxxCh. 4.5 - Differentiate the following functions. y=exlnxCh. 4.5 - Differentiate the following functions. y=e1+lnxCh. 4.5 - Differentiate the following functions. y=lnxxCh. 4.5 - Prob. 8ECh. 4.5 - Differentiate the following functions. y=lnx2Ch. 4.5 - Prob. 10ECh. 4.5 - Differentiate the following functions. y=ln(1x)Ch. 4.5 - Prob. 12ECh. 4.5 - Differentiate the following functions. y=ln(3x4x2)Ch. 4.5 - Prob. 14ECh. 4.5 - Differentiate the following functions. y=1lnxCh. 4.5 - Differentiate the following functions. y=lnxln2xCh. 4.5 - Differentiate the following functions. y=lnxln2xCh. 4.5 - Differentiate the following functions. y=(lnx)2Ch. 4.5 - Differentiate the following functions....Ch. 4.5 - Differentiate the following functions....Ch. 4.5 - Find the second derivatives. d2dt2(t2lnt)Ch. 4.5 - Find the second derivatives. d2dt2ln(lnt)Ch. 4.5 - The graph of f(x)=(lnx)/x is shown in Fig.4. Find...Ch. 4.5 - The graph of f(x)=x/(lnx+x) is shown in Fig.5....Ch. 4.5 - Write the equation of the tangent line to the...Ch. 4.5 - The function f(x)=(lnx+1)/x has a relative extreme...Ch. 4.5 - Determine the domain of definition of the given...Ch. 4.5 - Find the equations of the tangent lines to the...Ch. 4.5 - Find the coordinates of the relative extreme point...Ch. 4.5 - Repeat the previous exercise with y=xlnx.Ch. 4.5 - The graphs of y=x+lnx and y=ln2x are shown in...Ch. 4.5 - Prob. 32ECh. 4.5 - Prob. 33ECh. 4.5 - The function y=2x2ln4x (x0) has one minimum point....Ch. 4.5 - A Demand Equation If the demand equation for a...Ch. 4.5 - Total Revenue Suppose that the total revenue...Ch. 4.5 - An Area ProblemFind the maximum area of a...Ch. 4.5 - Analysis of the Effectiveness of an Insect...Ch. 4.6 - Differentiate f(x)=ln[ exx(x+1)6 ].Ch. 4.6 - Use logarithmic differentiation to differentiate...Ch. 4.6 - Simplify the following expressions. ln5+lnxCh. 4.6 - Simplify the following expressions. lnx5lnx3Ch. 4.6 - Simplify the following expressions. 12ln9Ch. 4.6 - Simplify the following expressions. 3ln12+ln16Ch. 4.6 - Simplify the following expressions. ln4+ln6ln12Ch. 4.6 - Simplify the following expressions. ln2lnx+ln3Ch. 4.6 - Simplify the following expressions. e2lnxCh. 4.6 - Simplify the following expressions. 32ln45ln2Ch. 4.6 - Simplify the following expressions. 5lnx12lny+3lnzCh. 4.6 - Simplify the following expressions. elnx2+3lnyCh. 4.6 - Simplify the following expressions. lnxlnx2+lnx4Ch. 4.6 - Prob. 12ECh. 4.6 - Simplify the following expressions. Which is...Ch. 4.6 - Simplify the following expressions. Which is...Ch. 4.6 - Evaluate the given expressions. Use ln2=.69 and...Ch. 4.6 - Evaluate the given expressions. Use ln2=.69 and...Ch. 4.6 - Evaluate the given expressions. Use ln2=.69 and...Ch. 4.6 - Prob. 18ECh. 4.6 - Which of the following is the same as 4ln2x? a....Ch. 4.6 - Prob. 20ECh. 4.6 - Which of the following is the same as ln8x2ln2x?...Ch. 4.6 - Which of the following is the same as ln9x2? a....Ch. 4.6 - Solve the given equation for x. lnxlnx2+ln3=0Ch. 4.6 - Solve the given equation for x. lnx2ln3=0Ch. 4.6 - Solve the given equation for x. lnx42lnx=1Ch. 4.6 - Solve the given equation for x. lnx2ln2x+1=0Ch. 4.6 - Solve the given equation for x. (lnx)21=0Ch. 4.6 - Solve the given equation for x. 3lnxln3x=0Ch. 4.6 - Solve the given equation for x. lnx=lnxCh. 4.6 - Solve the given equation for x. 2(lnx)2+lnx1=0Ch. 4.6 - Solve the given equation for x. ln(x+1)ln(x2)=1Ch. 4.6 - Solve the given equation for x....Ch. 4.6 - Differentiate. y=ln[(x+5)(2x1)(4x)]Ch. 4.6 - Differentiate. y=ln[(x+1)(2x+1)(3x+1)]Ch. 4.6 - Differentiate. y=ln[(1+x)2(2+x)3(3+x)4]Ch. 4.6 - Differentiate. y=ln[e2x(x3+1)(x4+5x)]Ch. 4.6 - Differentiate. y=ln[xex2+1]Ch. 4.6 - Prob. 38ECh. 4.6 - Differentiate. y=ln(x+1)4ex1Ch. 4.6 - Differentiate. y=ln(x+1)4(x3+2)x1Ch. 4.6 - Prob. 41ECh. 4.6 - Prob. 42ECh. 4.6 - Use logarithmic differentiation to differentiate...Ch. 4.6 - Use logarithmic differentiation to differentiate...Ch. 4.6 - Use logarithmic differentiation to differentiate...Ch. 4.6 - Use logarithmic differentiation to differentiate...Ch. 4.6 - Prob. 47ECh. 4.6 - Use logarithmic differentiation to differentiate...Ch. 4.6 - Use logarithmic differentiation to differentiate...Ch. 4.6 - Use logarithmic differentiation to differentiate...Ch. 4.6 - Prob. 51ECh. 4.6 - Prob. 52ECh. 4.6 - Prob. 53ECh. 4.6 - Prob. 54ECh. 4 - State as many laws of exponents as you can recall.Ch. 4 - Prob. 2CCECh. 4 - Prob. 3CCECh. 4 - Prob. 4CCECh. 4 - Prob. 5CCECh. 4 - Prob. 6CCECh. 4 - Prob. 7CCECh. 4 - Prob. 8CCECh. 4 - Prob. 9CCECh. 4 - Prob. 10CCECh. 4 - Prob. 11CCECh. 4 - Prob. 12CCECh. 4 - Prob. 13CCECh. 4 - Prob. 14CCECh. 4 - Calculate the following. 274/3Ch. 4 - Calculate the following. 41.5Ch. 4 - Prob. 3RECh. 4 - Prob. 4RECh. 4 - Calculate the following. (25/7)14/5Ch. 4 - Prob. 6RECh. 4 - Prob. 7RECh. 4 - Calculate the following. 40.240.3Ch. 4 - Simplify the following. (ex2)3Ch. 4 - Simplify the following. e5xe2xCh. 4 - Simplify the following. e3xexCh. 4 - Simplify the following. 2x3xCh. 4 - Simplify the following. (e8x+7e2x)e3xCh. 4 - Simplify the following. e5x/2e3xexCh. 4 - Solve the following equations for x. e3x=e12Ch. 4 - Solve the following equations for x. ex2x=e2Ch. 4 - Solve the following equations for x. (exe2)3=e9Ch. 4 - Solve the following equations for x. e5xe4=eCh. 4 - Differntiate the following functions. y=10e7xCh. 4 - Differntiate the following functions. y=exCh. 4 - Differentiate the following functions. y=xex2Ch. 4 - Differentiate the following functions. y=ex+1x1Ch. 4 - Differntiate the following functions. y=eexCh. 4 - Differntiate the following functions. y=(x+1)e2xCh. 4 - Differentiate the following functions....Ch. 4 - Differentiate the following functions. y=xeCh. 4 - The graph of the functions f(x)=ex24x2 is shown in...Ch. 4 - Show that the function in Fig. 1 has a relative...Ch. 4 - Solve the following equations for t....Ch. 4 - Solve the following equations for t. et8e0.02t=0Ch. 4 - Solve the equation 42x=ex. [Hint: Express 2x as an...Ch. 4 - Solve the equation 3x=2ex. [Hint: Express 3x as an...Ch. 4 - Find the points on the graph of y=ex where the...Ch. 4 - Find the points on the graph y=ex+e2x where the...Ch. 4 - Determine the intervals where the function...Ch. 4 - Determine the intervals where the function...Ch. 4 - Find the equation of the tangent line to the graph...Ch. 4 - Show that the tangent lines to the graph of...Ch. 4 - Simplify the following expressions. e(ln5)/2Ch. 4 - Simplify the following expressions. eln(x2)Ch. 4 - Simplify the following expressions. lnx2lnx3Ch. 4 - Simplify the following expressions. e2ln2Ch. 4 - Simplify the following expressions. e5ln1Ch. 4 - Simplify the following expressions. [elnx]2Ch. 4 - Solve the following equations for t. tlnt=eCh. 4 - Solve the following equations for t. ln(ln3t)=0Ch. 4 - Solve the following equations for t. 3e2t=15Ch. 4 - Solve the following equations for t. 3et/212=0Ch. 4 - Solve the following equations for t. 2lnt=5Ch. 4 - Solve the following equations for t. 2e0.3t=1Ch. 4 - Differentiate the following functions....Ch. 4 - Differentiate the following functions. y=xlnxCh. 4 - Differentiate the following functions. y=ln(5x7)Ch. 4 - Differentiate the following functions. y=ln(9x)Ch. 4 - Differentiate the following functions. y=(lnx)2Ch. 4 - Differentiate the following functions. y=(xlnx)3Ch. 4 - Differentiate the following functions....Ch. 4 - Differentiate the following functions....Ch. 4 - Differentiate the following functions. y=xlnxxCh. 4 - Differentiate the following functions. y=e2ln(x+1)Ch. 4 - Differentiate the following functions. y=ln(lnx)Ch. 4 - Differentiate the following functions. y=1lnxCh. 4 - Differentiate the following functions. y=exlnxCh. 4 - Differentiate the following functions. y=ln(x2+ex)Ch. 4 - Differentiate the following functions....Ch. 4 - Differentiate the following functions. y=ln|2x+1|Ch. 4 - Differentiate the following functions. y=ln(ex2x)Ch. 4 - Differentiate the following functions. y=lnx3+3x23Ch. 4 - Differentiate the following functions. y=ln(2x)Ch. 4 - Differentiate the following functions....Ch. 4 - Differentiate the following functions. y=ln|x1|Ch. 4 - Differentiate the following functions....Ch. 4 - Differentiate the following functions. y=ln(1ex)Ch. 4 - Differentiate the following functions....Ch. 4 - Use logarithmic differentiation to differentiate...Ch. 4 - Use logarithmic differentiation to differentiate...Ch. 4 - Use logarithmic differentiation to differentiate...Ch. 4 - Use logarithmic differentiation to differentiate...Ch. 4 - Use logarithmic differentiation to differentiate...Ch. 4 - Prob. 80RECh. 4 - Prob. 81RECh. 4 - Prob. 82RECh. 4 - Use logarithmic differentiation to differentiate...Ch. 4 - Prob. 84RECh. 4 - Prob. 85RECh. 4 - Prob. 86RECh. 4 - Prob. 87RECh. 4 - Health Expenditures The health expenditures (in...
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