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Chapter: 15 -Problem: 15 >> In Exercises 15–24, use Green’s Theorem to evaluate the line integral. Transcribed Image Text: 2xy dx + (x + y) dy C: boundary of the region lying between the graphs of y = 0 and y = 1 - x²
Answer Preview: To evaluate the line integral C 2xy dx x y dy using Green s Theorem we need to express the line integral as a double integral over the region enclosed …

, Chapter: 8 -Problem: 40 >> Use a table of integrals to find the indefinite integral. Transcribed Image Text: [ cot ? d?
Answer Preview: We can use trigonometric identities to express cot 4 in terms of other trigonometric functions U…

, Chapter: 9 -Problem: 52 >> Determine the degree of the Maclaurin polynomial required for the error in the approximation of the function at the indicated value of x to be less than 0.001.f(x) = cos x, approximate f(0.4)
Answer Preview: To determine the degree of the Maclaurin polynomial required for the error in the approximation of t…

, Chapter: 9 -Problem: 13 >> Use the Direct Comparison Test to determine the convergence or divergence of the series. Transcribed Image Text: 18 n = 0 - 2
Answer Preview: To apply the direct comparison test we need to compare the gi…

, Chapter: 13 -Problem: 16 >> In Exercises 9–20,evaluate the function at the given values of the independent variables. Simplify the results. Transcribed Image Text: V(r, h) = r²h (a) V(3, 10) (b) V(5,2) (c) V(4,8) (d) V(6, 7)
Answer Preview: To evaluate the function V r h rh at the given values of the in…

, Chapter: 10 -Problem: 71 >> Convert the rectangular equation to polar form and sketch its graph.-x + 4y - 3 = 0
Answer Preview: r 3 4…

, Chapter: 5 -Problem: 17 >> Sketch the graph of the function.y = (1/3)x
Answer Preview: 2 …

, Chapter: 8 -Problem: 99 >> The Gamma Function ?(n) is defined by (a) Find ?(1), ?(2), and ?(3). (b) Use integration by parts to show that ?(n + 1) = n?(n). (c) Write ?(n) using factorial notation where n is a positive integer. Transcribed Image Text: I(n) = *00 Jo 10 xn-le-dx,n>0.
Answer Preview: a Using the formula for the gamma function we have 1 x 1 1 e x dx e x dx 1 e 0 1 2 x 2 1 e x dx x e …

, Chapter: 6 -Problem: 51 >> The population (in millions) of a country in 2015 and the expected continuous annual rate of change k of the population are given.  (a) Find the exponential growth model P = Cekt for the population by letting t = 5 correspond to 2015. (b) Use the model to predict the population of the country in 2030. (c) Discuss the relationship between the sign of k and the change in population for the country.
Answer Preview: a P 2 113e 0 011…

, Chapter: 14 -Problem: 5 >> In Exercises 3–6, evaluate the iterated integral. Transcribed Image Text: S.S. Jo Jo T- 27 x³ dy dx
Answer Preview: ANSWER To evaluate the given iterated integral we start by integrating with respect to y first and t…

, Chapter: 13 -Problem: 70 >> A principal of $5000 is deposited in a savings account that earns interest at a rate of r (written as a decimal), compounded continuously. The amount A(r, t) after t years is A(r, t) = 5000ert. Use this function of two variables to complete the table. Transcribed Image Text: 0.02 Rate 5 0.03 0.04 N
Answer Preview: To complete the table using the function A r t 5000e rt we need to plug in different val…

, Chapter: 5 -Problem: 114 >> Evaluate the definite integral. Use a graphing utility to verify your result. Transcribed Image Text: Jo /2 xe-x/2 dx
Answer Preview: We can evaluate the definite integral using integration by substitution Let u x 2 2 then d…

, Chapter: 5 -Problem: 66 >> Find the area of the given region. Use a graphing utility to verify your result. Transcribed Image Text: y= -2 4e* 1 + ²x -I 3 -1 y x = ln ?3 1 2 X
Answer Preview: To find the area of the region under the curve y 4e x 1 e 2x from x ln sqrt 3 to infi…

, Chapter: 14 -Problem: 63 >> The value of the integral is required in the development of the normal probability density function. (a) Use polar coordinates to evaluate the improper integral. (b) Use the result of part (a) to determine I. Transcribed Image Text: roo = Sohbe 00 I e-²/2 dx
Answer Preview: ANSWER To evaluate the integral using polar coordinates we can express the integral as follows I e …

, Chapter: 2 -Problem: 52 >> Find the second derivative of the function.h(t) = ?12 csc t
Answer Preview: To find the second derivative of h t 12 csc t we ne…

, Chapter: 2 -Problem: 14 >> Use the rules of differentiation to find the derivative of the function.f(x) = x1/2 ? x?5/6
Answer Preview: To find the derivative of the function f x x 1 2 x 5 6 using …

, Chapter: 1 -Problem: 2 >> Discuss the relationship between secant lines through a fixed point and a corresponding tangent line at that fixed point.
Answer Preview: The relationship between secant lines and a tangent line at a fixed point on a curve is an important …

, Chapter: 1 -Problem: 81 >> Find the one-sided limit (if it exists). Transcribed Image Text: lim x-1+ X x + 1 3 x³ + 1
Answer Preview: To find the one sided limit as x approaches 1 fr…

, Chapter: 8 -Problem: 53 >> Find the arc length of the graph of the function over the given interval.y = ln x, [1, 5]
Answer Preview: To find the arc length of the graph of the function y ln x over the interval 1 5 we can …

, Chapter: 9 -Problem: 36 >> (a) Write the repeating decimal as a geometric series and (b) Write the sum of the series as the ratio of two integers.0.64?
Answer Preview: a To write the repeating decimal 0 64 as a geometric series we can represent it as follows 0 64 0 64…

, Chapter: 4 -Problem: 56 >> Find the indefinite integral by making a change of variables.? (x + 1) ?2 - x dx
Answer Preview: Let u 2 x then du dx 1 and dx du Substitut…

, Chapter: 13 -Problem: 18 >> In Exercises 9–20,evaluate the function at the given values of the independent variables. Simplify the results. Transcribed Image Text: = [² / dr dt g(x, y) = (a) g(4,1) (b) g(6, 3) (c) g(2,5) (d) g(1,7)
Answer Preview: ANSWER To evaluate the function g x y at the given …

, Chapter: 10 -Problem: 11 >> Find the area of the region.Two petals of r = sin 8?
Answer Preview: The polar curve r sin 8 has eight petals so to find the area of t…

, Chapter: 9 -Problem: 20 >> Verify that the infinite series converges. Transcribed Image Text: ?? ?² (-¹)" n=1
Answer Preview: We can use the ratio test to determine the convergence of this infin…

, Chapter: 5 -Problem: 10 >> Evaluate the expression without using a calculator.log27 1/9
Answer Preview: log base b of a c c log base b of a Usin…

, Chapter: 8 -Problem: 65 >> Find the general solution of the differential equation using any method. Transcribed Image Text: dy 25 2 dx x² - 25 ||
Answer Preview: Solve the differential equation using separation of variables y 25 x 5 x 5 Next separate the …

, Chapter: 6 -Problem: 43 >> Find the orthogonal trajectories for the family of curves. Use a graphing utility to graph several members of each family.3x2 - y2 = C
Answer Preview: Hyperbolas …

, Chapter: 14 -Problem: 62 >> In Exercises 61–64, evaluate the triple iterated integral. Transcribed Image Text: ?/2 /23 (4-z 0 z dr dz de
Answer Preview: ANSWER To evaluate the triple iterated integral z dr dz d where the limit…

, Chapter: 13 -Problem: 25 >> In Exercises 21–32, find the domain and range of the function. Transcribed Image Text: = 2 x + y xy
Answer Preview: To find the domain and range of the function z x y xy we need to consider the restrictions on the va…

, Chapter: 5 -Problem: 120 >> Find the general solution of the differential equation.dy/dx = (ex - e-x)2
Answer Preview: To solve this differential equation we can use separation of variables dy …

, Chapter: 5 -Problem: 61 >> Find an equation of the tangent line to the graph of the function at the given point.y = 2-x, (-1, 2)
Answer Preview: To find the equation of the tangent line to the graph of y 2 x at t…

, Chapter: 2 -Problem: 70 >> The relationship between f and g is given. Explain the relationship between f? and g?.g(x) = 2 f(x)
Answer Preview: If g x 2 f x then we can use the chain rule of differentiation …

, Chapter: 3 -Problem: 37 >> Find the absolute extrema of the function on the closed interval. Transcribed Image Text: f(x) = sin x, 5 11 6' 6
Answer Preview: Minim…

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, Chapter: 1 -Problem: 82 >> Find the one-sided limit (if it exists). Transcribed Image Text: x + 1 lim x--1-¹-1 X
Answer Preview: To find the one sided limit as x approaches 1 from the left i e x 1 we can simp…

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, Chapter: 8 -Problem: 32 >> Use substitution and partial fractions to find the indefinite integral. Transcribed Image Text: 1 x(??3-?x) - dx
Answer Preview: Let s start by simplifying the denominator by rationalizing it 1 x 3 x 1 x 3 x 3 …

, Chapter: 10 -Problem: 33 >> Find the standard form of the equation of the ellipse with the given characteristics.Vertices: (3, 1), (3, 9) Minor axis length: 6
Answer Preview: The center of the ellipse is the midpoint of the line segment connecting the vertic…

, Chapter: 4 -Problem: 16 >> Find the indefinite integral and check the result by differentiation. Transcribed Image Text: x(5x² + 4)³ dx
Answer Preview: ANSWER We can solve this integral using the substit…

, Chapter: 13 -Problem: 19 >> In Exercises 11–24, find the limit and discuss the continuity of the function. Transcribed Image Text: lim (x, y) (n/3, 2) y cos xy
Answer Preview: To find the limit and discuss the continuity of the function f x y ycos xy as x y approaches 3 2 we …

, Chapter: 10 -Problem: 20 >> Find the eccentricity and the distance from the pole to the directrix of the conic. Then identify the conic and sketch its graph. Use a graphing utility to confirm your results. Transcribed Image Text: -6 3 + 7 sin 0
Answer Preview: To compare the eccentricity of the given equation we compare the given equation r 6 3 7 sin with …

, Chapter: 9 -Problem: 41 >> Determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. Transcribed Image Text: an = 2¹/n
Answer Preview: To determine the convergence or divergence of the sequen…

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, Chapter: 8 -Problem: 89 >> Let f be continuous on the interval [a, ?). Show that if the improper integral  converges, then the improper integral also converges. Transcribed Image Text: xp |(x)f| D
Answer Preview: To show that if the improper integral of f x from a to converges then the improper integral of f x f…

, Chapter: 6 -Problem: 51 >> The rate of change of the number of raccoons N(t) in a population is directly proportional to 380 - N(t), where t is the time in years. When t = 0, the population is 110, and when t = 4, the population has increased to 150. Find the population when t = 8.
Answer Preview: Given information The rate of change of the number of raccoons N t in a population is directly propo…

, Chapter: 14 -Problem: 7 >> In Exercises 3–10, evaluate the triple iterated integral. Transcribed Image Text: SCC ² 10 0 2ze-x² dy dx dz
Answer Preview: To evaluate the triple iterated integral from 4 to 0 from 1 to 0 fr…

, Chapter: 13 -Problem: 42 >> Draw the level curves for f(x, y) = x2 + y2 = c for c = 1, 2, 3, and 4, and sketch the constraint x + y = 2. Explain analytically how you know that the extremum of f(x, y) = x2 + y2 at (1, 1) is a minimum instead of a maximum.
Answer Preview: To draw the level curves for the function f x y x 2 y 2 c where c 1 2 3 and 4 we can plot the curves …

, Chapter: 5 -Problem: 80 >> Evaluate the definite integral. Use a graphing utility to verify your result. Transcribed Image Text: 3 fi (4x+1 + 2x) dx
Answer Preview: To evaluate the definite integral 4 x 1 2 x dx from 1 to 3 we can use the following steps Firs…

, Chapter: 5 -Problem: 85 >> A meteorologist measures the atmospheric pressure P (in millibars) at altitude h (in kilometers). The data are shown below. (a) Use a graphing utility to plot the points (h, ln P). Use the regression capabilities of the graphing utility to find a linear model for the revised data points.(b) The line in part (a) has the form ln P = ah + b. Write the equation in exponential form.(c) Use a graphing
Answer Preview: a ln P 0 1499h 6 979…

, Chapter: 2 -Problem: 40 >> A balloon rises at a rate of 4 meters per second from a point on the ground 50 meters from an observer. Find the rate of change of the angle of elevation of the balloon from the observer when the balloon is 50 meters above the ground.
Answer Preview: Let s denote the distance between the balloon and the observer by d and the height of the balloon by …

, Chapter: 3 -Problem: 35 >> Find the absolute extrema of the function on the closed interval.f(x) = [[x]], [?2, 2]
Answer Preview: Minimum …

, Chapter: 1 -Problem: 35 >> Find the limit of the trigonometric function.lim x?3 tan x/4
Answer Preview: To find the limit of the trigonometric function lim x3 tan …

, Chapter: 15 -Problem: 19 >> In Exercises 19–22, evaluate the line integral along the given path. Transcribed Image Text: Lex C: r(t) = 4ti + 3tj 0 ?t?1 xy ds
Answer Preview: To evaluate the line integral C xy ds where C is the path defined by …

, Chapter: 8 -Problem: 25 >> Use a table of integrals to find the indefinite integral. Transcribed Image Text: ex arccos ex dx
Answer Preview: Using the table of integrals we can see that the given integral can be rewritten as 1 …

, Chapter: 9 -Problem: 23 >> Find the nth Maclaurin polynomial for the function. Transcribed Image Text: f(x) 1 1 - x' n = 5
Answer Preview: To find the nth Maclaurin polynomial for the function f …

, Chapter: 4 -Problem: 33 >> Find the general solution of the differential equation.dy/dx = x + 1/(x2 + 2x - 3)2
Answer Preview: To solve the differential equation we need to separate the variables x …

, Chapter: 13 -Problem: 11 >> In Exercises 9–20,evaluate the function at the given values of the independent variables. Simplify the results. Transcribed Image Text: f(x, y) = xex (a) f(-1,0) (b) f(0, 2) (c) f(x, 3) (d) f(t, -y)
Answer Preview: a f 1 0 As we have f x y xe y For 1 0 f 1 0 …

, Chapter: 10 -Problem: 73 >> Convert the polar equation to rectangular form and sketch its graph.r = 6 cos ?
Answer Preview: x y 6x …

, Chapter: 9 -Problem: 23 >> Verify that the infinite series converges. Transcribed Image Text: 18 n=1 1 n(n+1)
Answer Preview: The given series is sum n 1 to 1 n n 1 We can simplif…

, Chapter: 5 -Problem: 25 >> Solve for x.(a) x2 - x = log5 25 (b) 3x + 5 = log2 64
Answer Preview: a x 2 x log5 25 Using the definition of logarithms we know that log a b c if and only if a c b …

, Chapter: 8 -Problem: 20 >> Find the trigonometric integral. Transcribed Image Text: ?.?. sin². - dx 2
Answer Preview: We can start by using the trigonometric identity sin 2 …

, Chapter: 7 -Problem: 19 >> Sketch the region bounded by the graphs of the equations and find the area of the region.f(x) = 1/9x2, y = 1, x = 1, x = 2
Answer Preview: 17 …

, Chapter: 14 -Problem: 44 >> In Exercises 43–46, find the area of the surface given by z = f(x, y) that lies above the region R.f(x, y) = 8 + 4x - 5y R = {(x, y): x2 + y2 ? 1}
Answer Preview: To find the area of the surface given by z f x y that lies above the region R we need to integrate t…

, Chapter: 13 -Problem: 44 >> In Exercises 43–46, use a computer algebra system to graph the function. Transcribed Image Text: Z = z/144 - 16x² - 9y-
Answer Preview: Imagin…

, Chapter: 5 -Problem: 73 >> Find the derivative of the function.y = 2x sinh-1(2x) - ?1 + 4x2
Answer Preview: To find the derivative of the function y 2x sinh 2x 1 4x 2 we can …

, Chapter: 5 -Problem: 67 >> Find the area of the given region. Use a graphing utility to verify your result. y = csc(x + 1) Transcribed Image Text: -1 3- 1 1 2 X
Answer Preview: To find the area of the given region we need to integrate th…

, Chapter: 2 -Problem: 35 >> Describe the relationship between the rate of change of y and the rate of change of x in each expression. Assume all variables and derivatives are positive.(a) (b) Transcribed Image Text: dy dt = dx 3 dt
Answer Preview: a dy dt 3 dx dt means that y chan…

, Chapter: 1 -Problem: 12 >> Create a table of values for the function and use the result to determine whether f (x) approaches ? or ?? as x approaches ?3 from the left and from the right. Use a graphing utility to graph the function to confirm your answer. Transcribed Image Text: f(x) x² - 9
Answer Preview: To create a table of values for f x x x 2 9 we can choose some values of x and evaluate f x x f x 4 …

, Chapter: 15 -Problem: 61 >> In Exercises 61 and 62, use a line integral to find the area of the region R.R: triangle bounded by the graphs of y = 1/2 x, y = 6 - x, and y = x
Answer Preview: To find the area of the region R bounded by the graphs of y 1 2 x y 6 x and y x we can use a line i…

, Chapter: 8 -Problem: 47 >> Use a table of integrals to evaluate the definite integral. Transcribed Image Text: (?/2 t³ cos t dt 3
Answer Preview: Using the table of integrals we can rewrite the integrand as follows sin 2x dx …

, Chapter: 9 -Problem: 28 >> Find the sequence of partial sums S1, S2, S3, S4, and S5. Transcribed Image Text: ?7+1-17 + 1 49 1 343
Answer Preview: The given series can be written as 7 1 1 7 1 49 1 343 To find the sequence of …

, Chapter: 4 -Problem: 62 >> Evaluate the definite integral. Use a graphing utility to verify your result. Transcribed Image Text: x³(2x4 + 1)² dx
Answer Preview: ANSWER We can start by using the substitution u 2x 4 1 Then du …

, Chapter: 12 -Problem: 14 >> In Exercises 11–18, find r'(t).r(t) = 4?ti + t2 ?t j + ln t2k
Answer Preview: To find r t the derivative of the vector valued function r t 4ti …

, Chapter: 10 -Problem: 24 >> Find the standard form of the equation of the hyperbola with the given characteristics.Center: (0, 0)Vertex: (0, 3)Focus: (0, 6)
Answer Preview: Consider the given center h k 0 0 Vertex 0 2 So a 2 Focus …

, Chapter: 9 -Problem: 33 >> Find the sum of the convergent series. Transcribed Image Text: 8 + 6 +++
Answer Preview: To find the sum of the series we can analyze the pattern and use the formula for the sum of an in…

, Chapter: 6 -Problem: 63 >> The level of sound ? (in decibels) with an intensity of I is ?(I) = 10 log10(I/I0), where I0 is an intensity of 10-16 watt per square centimeter, corresponding roughly to the faintest sound that can be heard. Determine ?(I) for the following. (a) I = 10-14 watt per square centimeter (whisper) (b) I = 10-9 watt per square centimeter (busy street corner) (c) I = 10-4 watt per square centimeter (thre
Answer Preview: a 20…

, Chapter: 8 -Problem: 19 >> Find the trigonometric integral. Transcribed Image Text: [COS (TX cos³(x - 1) dx
Answer Preview: To solve this integral we can use the reduction formula for cosine pow…

, Chapter: 6 -Problem: 12 >> Find the general solution of the differential equation.xy' = y
Answer Preview: To solve the differential equation xy y we can rearrange it as y y 1 …

, Chapter: 15 -Problem: 51 >> In Exercises 49–52, demonstrate the property that ?c F · dr = 0 regardless of the initial and terminal points of C, where the tangent vector r'(t) is orthogonal to the force field F. Transcribed Image Text: F(x, y) = (x³ ? 2x¹³)i + (x ? 2)i C: r(t) = ti + 1²j
Answer Preview: To demonstrate the property that c F dr 0 we need to calculate the lin…

, Chapter: 13 -Problem: 47 >> Let F(u, v) be a function of two variables. Find a formula for f'(x) when(a) f(x) = F(4x, 4) and (b) f(x) = F(-2x, x2).
Answer Preview: a b F…

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, Chapter: 3 -Problem: 43 >> Determine whether the Mean Value Theorem can be applied to f on the closed interval [a, b]. If the Mean Value Theorem can be applied, find all values of c in the open interval (a, b) such thatIf the Mean Value Theorem cannot be applied, explain why not.f(x) = x + 2/x ? 1, [?3, 3] Transcribed Image Text:
Answer Preview: f is …

, Chapter: 1 -Problem: 45 >> Find the limit L. Then use the - definition to prove that the limit is L.lim x?4 (x + 2)
Answer Preview: step 1 To find the limit of the given function as x approaches 4 we need to e…

, Chapter: 15 -Problem: 36 >> In Exercises 35–38, evaluate ?C F · dr. Transcribed Image Text: F(x, y) = (x - y)i + (x + y)j C: r(t) = 4 cos ti + 3 sin tj, 0 ?t? 2?
Answer Preview: To evaluate the line integral C F dr where F x y z 2yz i z x j x y k an…

, Chapter: 8 -Problem: 1 >> Would you use numerical integration to evaluate  Explain. Transcribed Image Text: 2 (ex + 5x) dx? Jo
Answer Preview: Yes we would use numerical integration to evaluate the given integral The reason is that the integra…

, Chapter: 9 -Problem: 34 >> Find the interval of convergence of the power series. Transcribed Image Text: 18 n=1 n!x" (2n)!
Answer Preview: To find the interval of convergence of the power series n 1 to i…

, Chapter: 9 -Problem: 29 >> Use the Integral Test to determine the convergence or divergence of the p-series. Transcribed Image Text: 00 n=1 7
Answer Preview: Applying the Integral test on 1 n 7 n 1 Here f x 1 …

, Chapter: 4 -Problem: 25 >> Use upper and lower sums to approximate the area of the region using the given number of subintervals (of equal width.) y = 10/ x2 + 1 Transcribed Image Text: 10 8 6 4 2 y 1 2 -X
Answer Preview: ANSWER To approximate the area of the region under the curve y 10 x 2 1 using upper and lower sums w…

, Chapter: 12 -Problem: 11 >> In Exercises 9–14, find the unit tangent vector T(t) and a set of parametric equations for the tangent line to the space curve at point P.r(t) = cos ti + 3 sin t j + (3t - 4)k, P(1, 0, -4)
Answer Preview: To find the unit tangent vector and parametric equations for the tangent line to the space curve at …

, Chapter: 10 -Problem: 75 >> Convert the polar equation to rectangular form and sketch its graph.r = -4 sec ?
Answer Preview: x 4 5…

, Chapter: 9 -Problem: 66 >> A deposit of $100 is made in an account at the beginning of each month at an annual interest rate of 3% compounded monthly. The balance in the account after n months is An = 100(401)[(1.0025)n - 1].(a) Compute the first six terms of the sequence {An}.(b) Find the balance in the account after 5 years by computing the 60th term of the sequence.(c) Find the balance in the account after 20 years by co
Answer Preview: a To find the first six terms of the sequence An we plug in n 1 2 3 4 5 and 6 int…

, Chapter: 6 -Problem: 65 >> Show that ifthen Transcribed Image Text: y || 1 1+ be-kt
Answer Preview: To find the derivative of y with respect to t we can use the ch…

, Chapter: 8 -Problem: 42 >> Determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. Transcribed Image Text: 6 So ?/36 13 1 /36-x- dx
Answer Preview: To determine whether the improper integral converges or diverges we need to check if there are an…

, Chapter: 6 -Problem: 12 >> Find the general solution of the first-order linear differential equation for x > 0.(y - 1) sin x dx - dy = 0
Answer Preview: To solve the given differential equation we can rewrite it in the standard form of a first order …

, Chapter: 15 -Problem: 44 >> In Exercises 43–46, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. Transcribed Image Text: If F = yi + xj and C is given by r(t) = 4 sin ti + 3 cos tj for 0 ? t ? 7, then S JC F. dr = 0.
Answer Preview: ANSWER To determine whether the statement is true or false we need to calculate the line in…

, Chapter: 13 -Problem: 53 >> In Exercises 53–56, find the slopes of the surface in the x- and y directions at the given point. Transcribed Image Text: z = xy (1, 2, 2) -4 4 2 X
Answer Preview: To find the slopes of the surface in the x and y directions at the given poi…

, Chapter: 5 -Problem: 80 >> Perform the following steps to find the maximum area of the rectangle shown in the figure (a) Solve for c in the equation f (c) = f (c + x).(b) Use the result in part (a) to write the area A as a function of x.(c) Use a graphing utility to graph the area function. Use the graph to approximate the dimensions of the rectangle of maximum area. Determine the maximum area.(d) Use a graphing utility to
Answer Preview: a We are given f x 10xe x To find the value of c that maximizes the area of the rectangle we need to …

, Chapter: 2 -Problem: 88 >> Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If f(x) = ?g(x) + b, then f?(x) = ?g?(x).
Answer Preview: The statement is false To see why let s apply the de…

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, Chapter: 1 -Problem: 16 >> Find the limit.lim x?7 (x ? 4)3
Answer Preview: Using the limit rule for polynom…

, Chapter: 1 -Problem: 86 >> Find the one-sided limit (if it exists). Transcribed Image Text: lim x-0 sec ³ 3 2x
Answer Preview: To find the one sided limit as x approaches 0 from th…

, Chapter: 15 -Problem: 11 >> In Exercises 9–18, use the Divergence Theorem to evaluate and find the outward flux of F through the surface of the solid S bounded by the graphs of the equations. Use a computer algebra system to verify your results. Transcribed Image Text: SSF JsJ F. NdS
Answer Preview: ANSWER To evaluate the surface integral F dS and find the outward flux of F through the surface of t…

, Chapter: 8 -Problem: 15 >> Use partial fractions to find the indefinite integral. Transcribed Image Text: .6 – ?? 7x³ + x 3 dx
Answer Preview: To use partial fractions we need to write the fraction as a sum of simpler fractions 9 x …

, Chapter: 9 -Problem: 14 >> Confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the convergence or divergence of the series. Transcribed Image Text: 18 n=2 In n n³
Answer Preview: To use the Integral Test we need to verify that the series is positive decreasing and continuo…

, Chapter: 4 -Problem: 25 >> Find the indefinite integral and check the result by differentiation. Transcribed Image Text: x 1-x² dx
Answer Preview: ANSWER To integrate x sqrt 1 x 2 dx we can use the substi…

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, Chapter: 10 -Problem: 60 >> The polar coordinates of a point are given. Plot the point and find the corresponding rectangular coordinates for the point. Transcribed Image Text: (-6, 5? 6
Answer Preview: To plot the point 6 5 6 in polar coordinates and find its corresponding rectangul…

, Chapter: 9 -Problem: 57 >> Determine the convergence or divergence of the series. Transcribed Image Text: 18 n=1 arctan n
Answer Preview: We can use the ratio test to determine the convergence of t…

, Chapter: 5 -Problem: 25 >> Evaluate the limit, using L’Hôpital’s Rule if necessary. Transcribed Image Text: 7x³ lim 2x + 1 6x³ + 1
Answer Preview: We need to evaluate the limit Lim 7x 3 2x 1 6x 3 1 x By considering …

, Chapter: 8 -Problem: 40 >> Use partial fractions to find the indefinite integral. Transcribed Image Text: 4x - 2 3(x-1)² dx
Answer Preview: We need to decompose the given fraction into partial fractions of the form A …

, Chapter: 6 -Problem: 49 >> Find the general solution of the first-order differential equation for x > 0 by any appropriate method.dy/dx = e2x + y/ex - y
Answer Preview: To solve the given differential equation dy dx e 2x y e x y we …

, Chapter: 14 -Problem: 29 >> In Exercises 29 and 30, use a double integral to find the area of the shaded region. Transcribed Image Text: 2r=1- cos 3 2 -0
Answer Preview: ANSWER In Exercise 29 to find the area of the shaded …

, Chapter: 13 -Problem: 42 >> In Exercises 35–42, describe and sketch the surface given by the function. Transcribed Image Text: f(x, y) = Jxy, 0, x ? 0, y ? 0 x <0 or y < 0
Answer Preview: The given function is defined differently depending on the values of x and y Let s analyze the cases …

, Chapter: 5 -Problem: 96 >> Use the graph of f' shown in the figure to answer the following. (a) Approximate the slope of f at x = -1. Explain.(b) Approximate any open intervals on which the graph of f is increasing and any open intervals on which it is decreasing. Explain. Transcribed Image Text: -5-4 3 2 1 - 1 y -2 -3+ 1 -X
Answer Preview: a To approximate the slope of f at x 1 we need to look at the slope of the tangent line to …

, Chapter: 2 -Problem: 49 >> Find the second derivative of the function.f(?) = 3 tan ?
Answer Preview: 6 …

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, Chapter: 1 -Problem: 29 >> Find the limit of the trigonometric function.lim x? 1cos x/3
Answer Preview: To find the limit of cos x 3 as x approaches 1 we can s…

, Chapter: 1 -Problem: 88 >> Explain why the function has at least two zeros in the interval [1, 5].f(x) = 2 cos x
Answer Preview: The given function is f x 2 cos x First note that the co…

, Chapter: 15 -Problem: 11 >> In Exercises 11 and 12, use a computer algebra system to evaluate ?S? (x2 - 2xy) dS.S: z = 10 - x2 - y2, 0 ? x ? 2, 0 ? y ? 2
Answer Preview: The surface S is defined by the equation z 10 x 2 y 2 with the limits of integration 0 x 2 and 0 y 2 …

, Chapter: 8 -Problem: 57 >> Find the centroid of the region bounded by the graphs of the inequalities. Transcribed Image Text: y ? 3 ?x² +9 ??? y ? 0, x ? 4, x ? 4
Answer Preview: The region bounded by the inequalities is shown in the image below To find the centroid of this regi…

, Chapter: 4 -Problem: 58 >> Use the Second Fundamental Theorem of Calculus to find F'(x).F(x) = ?x1 1/t2 dt
Answer Preview: ANSWER To use the Second Fundamental Theorem of Calculus to find F x we need to first fi…

, Chapter: 11 -Problem: 28 >> (a) Find the projection of u onto v,(b) Find the vector component of u orthogonal to v.u = ?1, -1, 1?, v = ?2, 0, 2?
Answer Preview: To find the projection of vector u onto vector v we can use the formula …

, Chapter: 10 -Problem: 55 >> Use the integration capabilities of a graphing utility to approximate the area of the region bounded by the graph of the polar equation. Transcribed Image Text: r = 3 2 - cos 0
Answer Preview: To approximate the area of the region bounded by the polar equation we need to convert the polar equ…

, Chapter: 9 -Problem: 21 >> Confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the convergence or divergence of the series. Transcribed Image Text: 18 n=1 n nt +1
Answer Preview: Yes the Integral Test can be applied to the series Let f x x x …

, Chapter: 6 -Problem: 56 >> The number of bacteria in a culture is increasing according to the law of exponential growth. There are 125 bacteria in the culture after 2 hours and 350 bacteria after 4 hours. (a) Find the initial population. (b) Write an exponential growth model for the bacteria population. Let t represent the time in hours. (c) Use the model to determine the number of bacteria after 8 hours. (d) After how many
Answer Preview: We can use the formula for exponential growth to solve this problem The formula is N N0 e kt where N …

, Chapter: 8 -Problem: 56 >> Use integration tables to find or evaluate the integral. Transcribed Image Text: 1 1 + tan ?x dx
Answer Preview: To evaluate the integral 1 1 tan x dx we can use integration tables or apply a substitution Let s us…

, Chapter: 7 -Problem: 42 >> Sketch the region bounded by the graphs of the equations and find the area of the region.f(x) = -2x, g(x) = 1 - 3x
Answer Preview: To sketch the region bounded by the graphs of the equations and find its area we need to firs…

, Chapter: 14 -Problem: 3 >> In Exercises 3–8, evaluate the triple iterated integral. Transcribed Image Text: C?/2 1/2 (3 LSI -1/0 Jo r cos 0 dr de dz
Answer Preview: ANSWER To evaluate the triple iterated integral of the function f r z r cos where r range…

, Chapter: 13 -Problem: 39 >> In Exercises 35–42, describe and sketch the surface given by the function.z = -x2 - y2
Answer Preview: Par…

, Chapter: 5 -Problem: 106 >> The atmospheric pressure decreases with increasing altitude. At sea level, the average air pressure is one atmosphere (1.033227 kilograms per square centimeter). The table shows the pressures p (in atmospheres) at selected altitudes h (in kilometers). (a) Use a graphing utility to find a model of the form p = a + b ln h for the data. Explain why the result is an error message.(b) Use a graphing u
Answer Preview: a When trying to fit a model of the form p a b ln h to the given data using a graphing utility an error message may be obtained because the function i…

, Chapter: 2 -Problem: 60 >> Find the slope of the graph of the function at the given point. Use the derivative feature of a graphing utility to confirm your results. Transcribed Image Text: Function f(x) = tan x cotx Point (1, 1)
Answer Preview: To find the slope of the graph of the function at the given point …

, Chapter: 1 -Problem: 28 >> Use the graph to find the limit (if it exists). If the limit does not exist, explain why. Transcribed Image Text: lim tan x x- /2 ?? 2 1 y 2 . 3? 2 -x
Answer Preview: The limit of tan x as x approaches 2 from the left and right sides is not defined which means t…

, Chapter: 1 -Problem: 88 >> Find the one-sided limit (if it exists). Transcribed Image Text: lim X-0- 2 cos² X X
Answer Preview: To find the one sided limit as x approaches 0 from the lef…

, Chapter: 15 -Problem: 21 >> In Exercises 19–22, evaluate the line integral along the given path. Transcribed Image Text: [ (x² + y² + 2²) ds C: r(t) = sin ti + cos tj + 2k 0 ?t? ? 2
Answer Preview: To evaluate the line integral C x y z ds along the given pa…

, Chapter: 8 -Problem: 2 >> Why is completing the square useful when you are considering integration by trigonometric substitution?
Answer Preview: Completing the square is a useful technique in integration by trigonometric substitution because it …

, Chapter: 13 -Problem: 24 >> In Exercises 21–32, find the domain and range of the function. Transcribed Image Text: g(x, y) = y X
Answer Preview: To find the domain and range of the function g x y y x we …

, Chapter: 10 -Problem: 5 >> Consider the polar equation Use a graphing utility to graph the equation for e = 0.1,e = 0.25, e = 0.5, e = 0.75, and e = 0.9. Identify the conic and discuss the change in its shape as e = 1- and e = 0+. Transcribed Image Text: r= 4 1+ e sin 0
Answer Preview: 30 5 e 0 …

, Chapter: 9 -Problem: 76 >> Determine the convergence or divergence of the series. Transcribed Image Text: 18 n=1 -1 n 3 nº
Answer Preview: ANSWER To determine the convergence or divergence of the series we can us…

, Chapter: 5 -Problem: 16 >> Verify the identity. Transcribed Image Text: sinh? x = - 1 + cosh 2x 2
Answer Preview: We can use the identity cosh 2 x sinh 2 x 1 to derive the given identity Starting with the right …

, Chapter: 8 -Problem: 10 >> Use integration by parts to find the indefinite integral. Transcribed Image Text: x²ex/2 dx
Answer Preview: To integrate x 2 e x 2 dx using integration by parts we need to choose u and dv s…

, Chapter: 6 -Problem: 4 >> Although it is true for some functions f and g, a common mistake in calculus is to believe that the Product Rule for derivatives is (fg)' = f'g'. (a) Given g(x) = x, find f such that (fg)' = f'g'. (b) Given an arbitrary function g, find a function f such that (fg)' = f'g'. (c) Describe what happens if g(x) = ex.
Answer Preview: a If we take g x x then we need to find f x such that fg f g Using the product rule we have fg f g f…

, Chapter: 15 -Problem: 55 >> In Exercises 51–56, determine whether the vector field is conservative. If it is, find a potential function for the vector field. Transcribed Image Text: XZ F(x, y, z) = ³i – 5j + (5 ? 1)k - y
Answer Preview: To determine if the vector field F x y z z y i xy y 2 j x y 1 k is conserva…

, Chapter: 13 -Problem: 66 >> In Exercises 61–66, discuss the continuity of the function. Transcribed Image Text: f(x, y) = (sin(x² - y²) x² - y² x² = y² x² = y²
Answer Preview: ANSWER To discuss the continuity of the function f x y sin x 2 y 2 x 2 y 2 we need to …

, Chapter: 5 -Problem: 112 >> Evaluate the definite integral. Use a graphing utility to verify your result. Transcribed Image Text: ro J-2 x²ex²/2 dx
Answer Preview: Let u x 3 2 then du dx 3 2 x 2 and dx 2 3 u 2 3 du Substituting into the integ…

, Chapter: 2 -Problem: 78 >> Describe the x-values at which f is differentiable. Transcribed Image Text: f(x) -4 = r x2 – 4 - 5432 y 34 X
Answer Preview: We can determine the values of x for which f x x 2 x 2 4 is not differentiable by checking for any p…

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, Chapter: 1 -Problem: 44 >> Find the one-sided limit (if it exists). Transcribed Image Text: ( x ? ² / + 3) x limx x-0+
Answer Preview: ANSWER To find the one sided limit as x approaches 0 we need to evaluate the expres…

, Chapter: 15 -Problem: 8 >> In Exercises 7–10, determine whether the vector field is conservative. Transcribed Image Text: F(x, y) = y ln x¡ + (In x)² j x; i X
Answer Preview: To determine if the vector field F x y y ln x x i ln x 2 j is conservative we need t…

, Chapter: 8 -Problem: 19 >> Approximate the definite integral using the Trapezoidal Rule and Simpson’s Rule with n = 4. Compare these results with the approximation of the integral using a graphing utility. Transcribed Image Text: 4 ?xex dx
Answer Preview: We can use the Trapezoidal Rule and Simpson s Rule to approximate the definite integral of the given …

, Chapter: 13 -Problem: 131 >> Consider the function f(x, y) = (x2 + y2)2/3. Show that Transcribed Image Text: f(x, y)=3(x² 0, 4x + y²)1/3² (x, y) = (0, 0) (x, y) = (0, 0)
Answer Preview: To find the partial derivative f x x y we differentiate the function f x y x 2 y 2 2 3 with resp…

, Chapter: 9 -Problem: 67 >> The interval of convergence of the geometric series (a) Find the sum of the series when x = 5/2. Use a graphing utility to graph the first six terms of the sequence of partial sums and the horizontal line representing the sum of the series. (b) Repeat part (a) for x = -5/2. (c) Write a short paragraph comparing the rates of convergence of the partial sums with the sums of the series in parts(a) an
Answer Preview: a 8 3 b 8 13 c The alternating series converges mo…

, Chapter: 9 -Problem: 36 >> Determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. Transcribed Image Text: an
Answer Preview: We can see that the given series e n 4 n n 1 to n converges This is due to the fact that upon ap…

, Chapter: 5 -Problem: 1 >> Describe the meaning of arccos x in your own words.
Answer Preview: arccos x …

, Chapter: 8 -Problem: 3 >> Let I = ?40 f(x) dx, where f is shown in the figure. Let L(n) and R(n) represent the Riemann sums using the left-hand endpoints and right-hand endpoints of n subintervals of equal width. (Assume n is even.) Let T(n) and S(n) be the corresponding values of the Trapezoidal Rule and Simpson’s Rule. (a) For any n, list L(n), R(n), T(n), and I in increasing order.(b) Approximate S(4).
Answer Preview: a R n I T …

, Chapter: 6 -Problem: 3 >> Describe carrying capacity in your own words.
Answer Preview: The carrying ca…

, Chapter: 15 -Problem: 57 >> In Exercises 57–60, find the divergence of the vector field.F(x, y) = x2i + 2y2j
Answer Preview: To find the divergence …

, Chapter: 13 -Problem: 39 >> In Exercises 39–46, find the absolute extrema of the function over the region R. (In each case, R contains the boundaries.) Use a computer algebra system to confirm your results.f(x, y) = x2 - 4xy + 5R = {(x, y): 1 ? x ? 4, 0 ? y ? 2}
Answer Preview: To find the absolute extrema of the function f x y x 2 4xy 5 over the region R x y 1 x 4 0 y 2 we ne…

, Chapter: 5 -Problem: 83 >> Consider the function f(x) = xn, where n is odd. Does f-1 exist? Explain.
Answer Preview: Yes Functions of the form f …

, Chapter: 2 -Problem: 62 >> Find the slope of the graph of the function at the given point. Use the derivative feature of a graphing utility to confirm your results. Transcribed Image Text: Function f(x) = (sin x) (sin x + cos x) Point (41)
Answer Preview: To find the slope of the graph of the function at the given point we need to fin…

, Chapter: 2 -Problem: 13 >> Use the rules of differentiation to find the derivative of the function.h(x) = 6?x + 33?x
Answer Preview: 3…

, Chapter: 1 -Problem: 42 >> Find the limit L. Then find such that ?f(x) ? L? < whenever 0 < ?x ? c? < for (a) = 0.01 and (b) = 0.005.lim x?4 (x2 + 6)
Answer Preview: a To find the limit we simply plug in c 4 lim x4 x2 6 22 Now we need to find a value of such …

, Chapter: 1 -Problem: 70 >> Discuss the continuity of the composite function h(x) = f(g(x)).f(x) = sinxg (x) = x2
Answer Preview: To discuss the continuity of the composite function h x f g x we need to analyze the …

, Chapter: 15 -Problem: 13 >> In Exercises 13–16,(a) Find a piecewise smooth parametrization of the path C, and(b) Evaluate ?c (2x + 3?y) ds.C: line segments from (0, 0) to (1, 0) and (1, 0) to (2, 4)
Answer Preview: To find a piecewise smooth parametrization of the path C we need to break it down into two segments …

, Chapter: 8 -Problem: 24 >> Use a table of integrals to find the indefinite integral. Transcribed Image Text: 13 1 + sin 04 - de
Answer Preview: Unfortunately the given integral cannot be found in a standard …

, Chapter: 4 -Problem: 28 >> Find the upper and lower sums for the region bounded by the graph of the function and the x-axis on the given interval. Leave your answer in terms of n, the number of subintervals. Transcribed Image Text: Function f(x) = 7x² Interval [0, 3]
Answer Preview: To find the upper and lower sums of the function f x on the interval 0 3 with n subintervals we need …

, Chapter: 12 -Problem: 22 >> In Exercises 19–22, find the curvature of the curve, where s is the arc length parameter.r(s) = cos si + sin sj + 5k
Answer Preview: To find the curvature of the curve given by the vector valued function r s cos s i sin s j 5k where …

, Chapter: 10 -Problem: 36 >> Find two different sets of parametric equations for the rectangular equation.y = x2 - 3
Answer Preview: Set 1 Let s express the variable x in terms of a parameter t and then define the v…

, Chapter: 9 -Problem: 44 >> Determine the convergence or divergence of the series. Transcribed Image Text: 200 + + 208 216 224 + +
Answer Preview: The given series is an infinite series with the general term 1 8k 1…

, Chapter: 6 -Problem: 51 >> Find the general solution of the first-order differential equation for x > 0 by any appropriate method.y cos x - cos x + dy/dx = 0
Answer Preview: To solve the given differential equation y cos x co…

, Chapter: 8 -Problem: 18 >> Find the trigonometric integral. Transcribed Image Text: [sin² sin² x cos³x dx
Answer Preview: We can use integration by parts to solve this integral Let u sin …

, Chapter: 6 -Problem: 35 >> Find the general solution of the differential equation.dy/dx = 5x/y
Answer Preview: We can start solving the given differential equation by first separating the variables x and …

, Chapter: 14 -Problem: 63 >> In Exercises 61–64, evaluate the triple iterated integral. Transcribed Image Text: C C/2 (sin 8 B p² sin 0 cos 0 dp de do 10
Answer Preview: ANSWER To evaluate the triple iterated integral let s start by integrating with respect to then and …

, Chapter: 13 -Problem: 24 >> In Exercises 17–26,(a) Find an equation of the tangent plane to the surface at the given point and(b) Find a set of symmetric equations for the normal line to the surface at the given point.6xy = z, (-1, 1, -6)
Answer Preview: a To find an equation of the tangent plane to the surface 6xy z at the point 1 1 6 we can use the gr…

, Chapter: 5 -Problem: 100 >> Let  (a) Show that f is one-to-one if and only if bc - ad ? 0.(b) Given bc - ad ? 0, find f-1.(c) Determine the values of a, b, c, and d such that f = f-1. Transcribed Image Text: f(x) = = ax + b cx + d'
Answer Preview: ANSWER a To show that f is one to one if and only if bc ad 0 we need to prove two things i If bc ad 0 then f is one to one ii If f is one to one then bc ad 0 Proof i Suppose bc ad 0 We want to show th…

, Chapter: 2 -Problem: 70 >> Find the slope of the graph of the function at the given point. Use the derivative feature of a graphing utility to confirm your results. Transcribed Image Text: y X + COS X. l'a 2 al
Answer Preview: To find the slope of the graph of the function at the point 2 2 we …

, Chapter: 3 -Problem: 2 >> In your own words, describe the Mean Value Theorem.
Answer Preview: The Mean Value Theorem is a fundamental theorem in calculus that states that for any continu…

, Chapter: 1 -Problem: 4 >> Let P(3, 4) be a point on the circle x2 + y2 = 25 (see figure).(a) What is the slope of the line joining P and O(0, 0)?(b) Find an equation of the tangent line to the circle at P.(c) Let Q(x, y) be another point on the circle in the first quadrant. Find the slope mx of the line joining P and Q in terms of x.(d) Calculate lim x?3 mx. How does this number relate to your answer in part (b)? 
Answer Preview: a The slope of the line joining P 3 4 and O 0 0 is m y2 y1 x2 x1 0 4 0 3 4 3 b To find the equation …

, Chapter: 1 -Problem: 49 >> Find the one-sided limit (if it exists).lim x?(1/2)? x sec x
Answer Preview: To find the one sided limit of the function f x x sec x as x approaches 1 2 from the le…

, Chapter: 15 -Problem: 12 >> In Exercises 11 and 12, use a computer algebra system to evaluate ?S? (x2 - 2xy) dS. Transcribed Image Text: 1 S: z = cos x, 0?x?7, 0? y ? x
Answer Preview: To evaluate the double integral S x 2 2xy dS over the su…

, Chapter: 8 -Problem: 18 >> Approximate the definite integral using the Trapezoidal Rule and Simpson’s Rule with n = 4. Compare these results with the approximation of the integral using a graphing utility. Transcribed Image Text: 2 1 /1 + x³ dx
Answer Preview: Using the Trapezoidal Rule with n 4 we have x 5 1 4 1 f 1 3 01 …

, Chapter: 4 -Problem: 14 >> Find the indefinite integral and check the result by differentiation. Transcribed Image Text: [x²(6 - x³)³ dx
Answer Preview: We can solve the integral by using the substitution …

, Chapter: 12 -Problem: 5 >> Find the domain of the vector-valued function.r(t) = ln ti - etj - tk
Answer Preview: To find the domain of the vector valued function r t ln t i e tj tk we need to deter…

, Chapter: 10 -Problem: 22 >> Find the standard form of the equation of the hyperbola with the given characteristics.Vertices: (±2,0) Asymptotes: y = ±32x
Answer Preview: To find the standard form of the equation of a hyperbola with the given characteristics we start wit…

, Chapter: 9 -Problem: 19 >> Confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the convergence or divergence of the series. Transcribed Image Text: 18 n=1 An 2n² + 1
Answer Preview: Applying the Integral test on 4n 2n 2 1 n 1 Here f x 4x 2…

, Chapter: 8 -Problem: 41 >> Determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. Transcribed Image Text: Sexte 4 2 /2x²??/x² - 4 J2 dx
Answer Preview: First let s check if the integral converges or diverges …

, Chapter: 6 -Problem: 10 >> Find the general solut

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