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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Chapter 4.1, Problem 30E
To determine
To calculate: The value of
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Provethat
a) prove that for any irrational numbers there exists?
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8. Evaluate arctan x dx
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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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- 12. Evaluate ſ √9-x2 -dx. x2 a) C 9-x2 √9-x2 - x2 b) C - x x arcsin ½-½ c) C + √9 - x² + arcsin x d) C + √9-x2 x2 13. Find the indefinite integral S cos³30 √sin 30 dᎾ . 2√√sin 30 (5+sin²30) √sin 30 (3+sin²30) a) C+ √sin 30(5-sin²30) b) C + c) C + 5 5 5 10 d) C + 2√√sin 30 (3-sin²30) 2√√sin 30 (5-sin²30) e) C + 5 15 14. Find the indefinite integral ( sin³ 4xcos 44xdx. a) C+ (7-5cos24x)cos54x b) C (7-5cos24x)cos54x (7-5cos24x)cos54x - 140 c) C - 120 140 d) C+ (7-5cos24x)cos54x e) C (7-5cos24x)cos54x 4 4 15. Find the indefinite integral S 2x2 dx. ex - a) C+ (x²+2x+2)ex b) C (x² + 2x + 2)e-* d) C2(x²+2x+2)e¯* e) C + 2(x² + 2x + 2)e¯* - c) C2x(x²+2x+2)e¯*arrow_forward4. Which substitution would you use to simplify the following integrand? S a) x = sin b) x = 2 tan 0 c) x = 2 sec 3√√3 3 x3 5. After making the substitution x = = tan 0, the definite integral 2 2 3 a) ៖ ស្លឺ sin s π - dᎾ 16 0 cos20 b) 2/4 10 cos 20 π sin30 6 - dᎾ c) Π 1 cos³0 3 · de 16 0 sin20 1 x²√x²+4 3 (4x²+9)2 π d) cos²8 16 0 sin³0 dx d) x = tan 0 dx simplifies to: de 6. In order to evaluate (tan 5xsec7xdx, which would be the most appropriate strategy? a) Separate a sec²x factor b) Separate a tan²x factor c) Separate a tan xsecx factor 7. Evaluate 3x x+4 - dx 1 a) 3x+41nx + 4 + C b) 31n|x + 4 + C c) 3 ln x + 4+ C d) 3x - 12 In|x + 4| + C x+4arrow_forward1. Abel's Theorem. The goal in this problem is to prove Abel's theorem by following a series of steps (each step must be justified). Theorem 0.1 (Abel's Theorem). If y1 and y2 are solutions of the differential equation y" + p(t) y′ + q(t) y = 0, where p and q are continuous on an open interval, then the Wronskian is given by W (¥1, v2)(t) = c exp(− [p(t) dt), where C is a constant that does not depend on t. Moreover, either W (y1, y2)(t) = 0 for every t in I or W (y1, y2)(t) = 0 for every t in I. 1. (a) From the two equations (which follow from the hypotheses), show that y" + p(t) y₁ + q(t) y₁ = 0 and y½ + p(t) y2 + q(t) y2 = 0, 2. (b) Observe that Hence, conclude that (YY2 - Y1 y2) + P(t) (y₁ Y2 - Y1 Y2) = 0. W'(y1, y2)(t) = yY2 - Y1 y2- W' + p(t) W = 0. 3. (c) Use the result from the previous step to complete the proof of the theorem.arrow_forward
- 2. Observations on the Wronskian. Suppose the functions y₁ and y2 are solutions to the differential equation p(x)y" + q(x)y' + r(x) y = 0 on an open interval I. 1. (a) Prove that if y₁ and y2 both vanish at the same point in I, then y₁ and y2 cannot form a fundamental set of solutions. 2. (b) Prove that if y₁ and y2 both attain a maximum or minimum at the same point in I, then y₁ and Y2 cannot form a fundamental set of solutions. 3. (c) show that the functions & and t² are linearly independent on the interval (−1, 1). Verify that both are solutions to the differential equation t² y″ – 2ty' + 2y = 0. Then justify why this does not contradict Abel's theorem. 4. (d) What can you conclude about the possibility that t and t² are solutions to the differential equation y" + q(x) y′ + r(x)y = 0?arrow_forwardQuestion 4 Find an equation of (a) The plane through the point (2, 0, 1) and perpendicular to the line x = y=2-t, z=3+4t. 3t, (b) The plane through the point (3, −2, 8) and parallel to the plane z = x+y. (c) The plane that contains the line x = 1+t, y = 2 − t, z = 4 - 3t and is parallel to the plane 5x + 2y + z = 1. (d) The plane that passes through the point (1,2,3) and contains the line x = 3t, y = 1+t, and z = 2-t. (e) The plane that contains the lines L₁: x = 1 + t, y = 1 − t, z = 2t and L2 : x = 2 − s, y = s, z = 2.arrow_forwardPlease find all values of x.arrow_forward
- 3. Consider the initial value problem 9y" +12y' + 4y = 0, y(0) = a>0: y′(0) = −1. Solve the problem and find the value of a such that the solution of the initial value problem is always positive.arrow_forward5. Euler's equation. Determine the values of a for which all solutions of the equation 5 x²y" + axy' + y = 0 that have the form (A + B log x) x* or Ax¹¹ + Bä” tend to zero as a approaches 0.arrow_forward4. Problem on variable change. The purpose of this problem is to perform an appropriate change of variables in order to reduce the problem to a second-order equation with constant coefficients. ty" + (t² − 1)y'′ + t³y = 0, 0arrow_forward4. Some psychologists contend that the number of facts of a certain type that are remembered after t hours is given by f(t)== 90t 951-90 Find the rate at which the number of facts remembered is changing after 1 hour and after 10 hours. Interpret.arrow_forward12:05 MA S 58 58. If f(x) = ci.metaproxy.org 25 2xon [0, 10] and n is a positive integer, then there is some Riemann sum Sthat equals the exact area under the graph of ƒ from x = Oto x = 10. 59. If the area under the graph of fon [a, b] is equal to both the left sum L, and the right sum Rfor some positive integer n, then fis constant on [a, b]. 60. If ƒ is a decreasing function on [a, b], then the area under the graph of fis greater than the left sum Land less than the right sum R₂, for any positive integer n. Problems 61 and 62 refer to the following figure showing two parcels of land along a river: River Parcel 2 Parcel 1 h(x) 500 ft 1,000 ft. Figure for 61 and 62 61. You want to purchase both parcels of land shown in the figure and make a quick check on their combined area. There is no equation for the river frontage, so you use the average of the left and right sums of rectangles covering the area. The 1,000-foot baseline is divided into 10 equal parts. At the end of each…arrow_forwardIf a snowball melts so that its surface area decreases at a rate of 10 cm²/min, find the rate (in cm/min) at which the diameter decreases when the diameter is 12 cm. (Round your answer to three decimal places.) cm/minarrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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