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Chapter: 4 -Problem: 70 >> Let ƒ be a function defined on an interval [a, b]. What conditions could you place on ƒ to guarantee that where min ƒ? and max ƒ? refer to the minimum and maximum values of ƒ? on [a, b]? Give reasons for your answers. Transcribed Image Text: min f' ? f(b) f(a) b - a < max f',
Answer Preview: Answer To guarantee that the inequality max x min x 2 b a b a holds for a function defined on an int…

, Chapter: 4 -Problem: 31 >> Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. Transcribed Image Text: y || X 2 ?x² + 1
Answer Preview: 4 3 2 …

, Chapter: 4 -Problem: 43 >> Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. Transcribed Image Text: (-2 cos t) dt
Answer Preview: ANSWER To find the antiderivative of 2cos t dt we can use the trigonometric identity …

, Chapter: 16 -Problem: 10 >> Find a parametrization of the surface.The surface cut from the parabolic cylinder y = x2 by the planes z = 0, z = 3, and y = 2
Answer Preview: To find a parametrization of the surface cut from the parabolic cylinder y x 2 by the planes z 0 z 3 …

, Chapter: 5 -Problem: 19 >> Oil is leaking out of a tanker damaged at sea. The damage to the tanker is worsening as evidenced by the increased leakage each hour, recorded in the following table. a. Give an upper and a lower estimate of the total quantity of oil that has escaped after 5 hours. b. Repeat part (a) for the quantity of oil that has escaped after 8 hours. c. The tanker continues to leak 720 gal / h after the first
Answer Preview: a To give an upper and a lower estimate of the total quantity of oil that has escaped after 5 hours …

, Chapter: 16 -Problem: 22 >> Apply Green’s Theorem to evaluate the integrals. Transcribed Image Text: $ C C: The boundary of 0 ? x ? ?, 0 ? y ? sin x (3y dx + 2x dy)
Answer Preview: To apply Green s Theorem we need to find a vector field whose curl is the given vector …

, Chapter: 14 -Problem: 11 >> (a) Express ?u/?x, ?u/?y, and ?u/?z as functions of x, y, and z both by using the Chain Rule and by expressing u directly in terms of x, y, and z before differentiating. Then  (b) Evaluate ?u/?x, ?u/?y, and ?u/?z at the given point (x, y, z). Transcribed Image Text: P - q q r r = x + y U = - - p = x
Answer Preview: a To express u x u y and u z using the Chain Rule you can use the following for…

, Chapter: 12 -Problem: 40 >> Write inequalities to describe the set.The closed region bounded by the spheres of radius 1 and radius 2 centered at the origin.
Answer Preview: The set can be described as the solid region between two concentric sph…

, Chapter: 4 -Problem: 104 >> Sketch a smooth connected curve y = ƒ(x) with Transcribed Image Text: f(-2) = 8, f(0) = 4, f(2)= 0, f'(x) > 0 for |x| > 2, f'(2) = f'(-2) = 0, f'(x) < 0 for x| < 2, f"(x) < 0 for x < 0, f"(x) > 0 for x > 0.
Answer Preview: ANSWER One possible curve that satisfies all the given conditions is as fol…

, Chapter: 3 -Problem: 42 >> Find the derivatives of the function.y = ?2e?2x
Answer Preview: To find the derivative of y with respect to x we can use the c…

, Chapter: 8 -Problem: 75 >> Another way to integrate ƒ-1(x) (when ƒ-1 is integrable, of course) is to use integration by parts with u = ƒ-1(x) and d? = dx to rewrite the integral of ƒ-1 as compare the results of using Equations (4) and (5). Equations (4) and (5) give different formulas for the integral of cos-1 x: a. b. Transcribed Image Text:
Answer Preview: ANSWER Let s simplify Equations 4 and 5 and replace sin 1 x wi…

, Chapter: 15 -Problem: 12 >> Let D be the region bounded below by the cone z = ?x2 + y2 and above by the paraboloid z = 2 - x2 - y2. Set up the triple integrals in cylindrical coordinates that give the volume of D using the following orders of integration.a. dz dr d? b. dr dz d? c. d? dz dr
Answer Preview: To set up the triple integrals in cylindrical coordinates we need to express the equations of the co…

, Chapter: 12 -Problem: 26 >> Sketch the surfaces.4x2 + 9z2 = 9y2
Answer Preview: The surface o…

, Chapter: 5 -Problem: 90 >> If your CAS can draw rectangles associated with Riemann sums, use it to draw rectangles associated with Riemann sums that converge to the integral. Use n = 4, 10, 20, and 50 subintervals of equal length in each case. Transcribed Image Text: L'ox² Jo (x² + 1) dx = 3
Answer Preview: ANSWER To draw the rectangles associated with Riemann sums that converge to the integral of f x x 2 …

, Chapter: 14 -Problem: 18 >> Find parametric equations for the line tangent to the curve of intersection of the surfaces at the given point.Surfaces: x2 + y2 = 4, x2 + y2 - z = 0Point: (?2, ?2, 4)
Answer Preview: To find the parametric equations for the line tangent to the curve of intersection of the surfaces a…

, Chapter: 11 -Problem: 16 >> Find the lengths of the curves.x = (y3/12) + (1/y), 1 ? y ? 2
Answer Preview: We are given the equation of the curve as x y 3 12 1 y 1 y 2 To find the length of this c…

, Chapter: 4 -Problem: 8 >> Find the absolute maximum and absolute minimum values of ƒ over the interval.ƒ(x) = (4/x) + ln x2, 1 ? x ? 4
Answer Preview: ANSWER First we need to find the critical points of in the interval 1 …

, Chapter: 11 -Problem: 2 >> Find the areas of the region. Bounded by the circle r = 2 sin u for ?/4 ? ? ? ?/2 Transcribed Image Text: r = 2 sin 0 0 (?) 2 4 X
Answer Preview: We can use polar coordinates to find the area of the region bounded by th…

, Chapter: 11 -Problem: 51 >> Find a polar equation in the form r cos (? - ?0) = r0 for each of the lines.y = -5
Answer Preview: To convert the given equation of a line in Cartesian coordinates into a polar …

, Chapter: 9 -Problem: 11 >> The autonomous differential equations represent models for population growth. For each exercise, use a phase line analysis to sketch solution curves for P(t), selecting different starting values P(0). Which equilibria are stable, and which are unstable? Transcribed Image Text: / dP dt = 2P(P - 3)
Answer Preview: To analyze the given differential equation using a phase line we need to determine the equilibrium s…

, Chapter: 8 -Problem: 18 >> Estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10-4 by  (a) The Trapezoidal Rule and  (b) Simpson’s Rule. Transcribed Image Text: 1 ds (s 1)² 2
Answer Preview: To estimate the minimum number of subintervals needed to approximate the integral 2 to 4 1 s 1 2 ds …

, Chapter: 10 -Problem: 27 >> Which of the series converge absolutely, which converge, and which diverge? Give reasons for your answers. Transcribed Image Text: ?(-1)"n?(2/3)" _n=1
Answer Preview: ANSWER To determine the convergence or divergence of the giv…

, Chapter: 12 -Problem: 4 >> The line through the origin and the point A(1, 1, 1) is the axis of rotation of a rigid body rotating with a constant angular speed of 3 / 2 rad / sec. The rotation appears to be clockwise when we look toward the origin from A. Find the velocity v of the point of the body that is at the position B(1, 3, 2). Transcribed Ima
Answer Preview: To find the velocity vector of the point B 1 3 2 we need to find its velocity components along the tangent and normal vectors of the rotation axis Fir…

, Chapter: 1 -Problem: 36 >> Use the addition formulas to derive the identities.sin (A - B) = sin Acos B - cos Asin B
Answer Preview: To derive the identity sin A B sin A cos B cos A sin B using th…

, Chapter: 1 -Problem: 13 >> Copy and complete the following table. Transcribed Image Text: g(x) a. X 7 b. x + 2 c. ? d. X x - 1 e. ? 1 X f. f(x) Vx 3x ?x - 5 X X - 1 + ? 1 X (fog)(x) ? ? ?x²-5 ? X X
Answer Preview: g x a x 7 b x 2 c d …

, Chapter: 4 -Problem: 58 >> a. Find the local extrema of each function on the given interval, and say where they occur. b. Graph the function and its derivative together. Comment on the behavior of ƒ in relation to the signs and values of ƒ?. Transcribed Image Text: f(x) = sin x - COS X, 0? x ? 2T
Answer Preview: ANSWER a To find the local extrema of the function f x sin x cos x on the interval 0 2 we first find …

, Chapter: 4 -Problem: 4 >> Identify the inflection points and local maxima and minima of the functions graphed. Identify the intervals on which the functions are concave up and concave down. Transcribed Image Text: 0 X< A y (L = 2xx =
Answer Preview: To find the inflection points and local maxima and minima we need to take the first and …

, Chapter: 4 -Problem: 75 >> Verify the formula differentiation. Transcribed Image Text: 1 1?x + ?²x = -x + ? + C 1 dx (x 1)² 1
Answer Preview: ANSWER To verify the given formula we will differentiate the right hand …

, Chapter: 15 -Problem: 9 >> Sketch the region bounded by the given lines and curves. Then express the region’s area as an iterated double integral and evaluate the integral.The lines y = x, y = x/3, and y = 2
Answer Preview: To sketch the region bounded by the given lines and curves we first plot the three lines y x y x 3 …

, Chapter: 16 -Problem: 12 >> Find a potential function ƒ for the field F. Transcribed Image Text: F = y 1 + x²y² · (²²2² = 1^ + ²²²x + 1). i+ + y (V-2 1- y² 7² + ½ ) K k
Answer Preview: To find a potential function for the vector field F we nee…

, Chapter: 16 -Problem: 23 >> Use a parametrization to express the area of the surface as a double integral. Then evaluate the integral.The cap cut from the paraboloid z = 2 - x2 - y2 by the cone z = ?x2 + y2
Answer Preview: To express the area of the surface as a double integral we first need to parameterize the surface …

, Chapter: 13 -Problem: 5 >> Give the position vectors of particles moving along various curves in the xy-plane. In each case, find the particle’s velocity and acceleration vectors at the stated times and sketch them as vectors on the curve.x2 + y2 = 1r(t) = (sin t)i + (cos t)j; t = ?/4 and ?/2
Answer Preview: The curve given by x 2 y 2 1 is a circle of radius 1 centered at the origin in the xy plane The posi…

, Chapter: 5 -Problem: 1 >> Write the sums without sigma notation. Then evaluate them. Transcribed Image Text: 2 k=1 6k k + 1
Answer Preview: To evaluate the sum 6k k 1 where k ranges from 1 to 2 we …

, Chapter: 3 -Problem: 90 >> Find dy/dt when x = 1 if y = x2 + 7x - 5 and dx/dt = 1/3.
Answer Preview: y x 2 7x 5 dx dt 1 3 We need to find dy dt when x 1 To find dy dt we …

, Chapter: 8 -Problem: 70 >> Use integration by parts to obtain the formula Transcribed Image Text: | - [ V? = x² dx = 1/2 x VI = x² + 1/ / 2 1 V1- ?T-1² dx.
Answer Preview: ANSWER To integrate the given function we will use integratio…

, Chapter: 15 -Problem: 12 >> Sketch the region bounded by the given lines and curves. Then express the region’s area as an iterated double integral and evaluate the integral.The lines y = x - 2 and y = -x and the curve y = ?x
Answer Preview: To sketch the region bounded by the given lines and curves we first plot the lines and the curve on …

, Chapter: 12 -Problem: 33 >> Find the distance from the point to the line.(0, 0, 12); x = 4t, y = -2t, z = 2t
Answer Preview: To find the distance from the point 0 0 12 to the line given by x 4t y 2t z 2t we can use the formul…

, Chapter: 5 -Problem: 85 >> Let F(x) = ?axƒ(t) dt for the specified function ƒ and interval [a, b]. Use a CAS to perform the following steps and answer the questions posed.a. Plot the functions ƒ and F together over [a, b].b. Solve the equation F?(x) = 0. What can you see to be true about the graphs of ƒ and F at points where F?(x) = 0? Is your observation borne out by Part 1 of the Fundamental Theorem coupled with informati
Answer Preview: a To plot and F together over a b we first need to calculate F x Using the given formula we have F x ax t dt 0x t 3 4t 2 3t dt t 4 4 4t 3 3 3t 2 2 lim…

, Chapter: 14 -Problem: 20 >> A particle traveling in a straight line with constant velocity i + j - 5k passes through the point (0, 0, 30) and hits the surface z = 2x2 + 3y2. The particle ricochets off the surface, the angle of reflection being equal to the angle of incidence. Assuming no loss of speed, what is the velocity of the particle after the ricochet? Simplify your answer.
Answer Preview: The velocity vector of the particle is given by v i j 5k We can first find the point on the surface z 2x 2 3y 2 where the particle hits by solving for …

, Chapter: 11 -Problem: 18 >> Graph the sets of points whose polar coordinates satisfy the equations and inequalitie.? = 11?/4, r ? -1
Answer Preview: The polar equation is given as follows 11 4 and r 1 To graph this we star…

, Chapter: 4 -Problem: 12 >> To calculate a planet’s space coordinates, we have to solve equations like x = 1 + 0.5 sin x. Graphing the function ƒ(x) = x - 1 - 0.5 sin x suggests that the function has a root near x = 1.5. Use one application of Newton’s method to improve this estimate. That is, start with x0 = 1.5 and find x1. (The value of the root is 1.49870 to five decimal places.) Remember to use radians.
Answer Preview: To use Newton s method we start with an initial guess x0 and itera…

, Chapter: 11 -Problem: 3 >> Plot the following points (given in polar coordinates). Then find all the polar coordinates of each point.a. (2, ?/2) b. (2, 0)c. (-2, ?/2) d. (-2, 0)
Answer Preview: a 2 2 represents a point which is 2 units away from the origin at an angle of 2 radians from the positive x axis in the upward direction To plot this …

, Chapter: 11 -Problem: 58 >> Find the center, foci, vertices, asymptotes, and radius, as appropriate, of the conic section.2x2 + 2y2 - 28x + 12y + 114 = 0
Answer Preview: We can begin by putting the given equation in standard form for a conic section which is x h a y k b …

, Chapter: 10 -Problem: 91 >> Suppose that a1, a2, a3,· · ·, an are positive numbers satisfying the following conditions: i) a1 ? a2 ? a3 ? · · ·; ii) The series a2 + a4 + a8 + a16 + · · · diverges. Show that the series diverges. Transcribed Image Text: a1 ?? + 1 2 + ?? 3 +
Answer Preview: ANSWER Since the series a2 a4 a8 a16 diverges we know that …

, Chapter: 8 -Problem: 31 >> If ƒ is continuously differentiable on [0, 1] and ƒ(1) = ƒ(0) = -1/6, prove that Transcribed Image Text: ] 0 2 / Fox (f'(x))² dx ? 2 f(x) dx + 1/ 4
Answer Preview: We can start the proof by using integration by parts on the left hand side of the inequality 0 1 f x …

, Chapter: 10 -Problem: 28 >> Find a polynomial that will approximate F(x) throughout the given interval with an error of magnitude less than 10-3. (a) [0, 0.5]  (b) [0, 1] Transcribed Image Text: F(x) = 0 In (1 + 1) t -dt,
Answer Preview: To find a polynomial that will approximate the function F x from x to 0 ln 1 t t dt with an error of magnitude less than 10 3 we can use Taylor s theo…

, Chapter: 5 -Problem: 67 >> If you do not know what substitution to make, try reducing the integral step by step, using a trial substitution to simplify the integral a bit and then another to simplify it some more. You will see what we mean if you try the sequences of substitution. a. u = tan x, followed by y = v3, then by w = 2 + v b. u = tan3 x, followed by v = 2 + u c. u = 2 + tan3 x
Answer Preview: ANSWER 1 To deduce whether the infinite series u1 u2 u3 is convergent we need to look at the behavio…

, Chapter: 12 -Problem: 6 >> Match the equation with the surface it defines. Also, identify each surface by type (paraboloid, ellipsoid, etc.). The surfaces are labeled (a)–(1). x = -y2 - z2 Transcribed Image Text: a. C. e. g. i. k. X X Z NE b. d. f. h. j. 1. X y
Answer Preview: As we have given with the equation x y 2 z 2 whose conic secti…

, Chapter: 1 -Problem: 32 >> Find the (a) Domain and (b) Range. Transcribed Image Text: y = -x - 2, X, -x + 2, -2 ? x ?-1 -1 < x < 1 1 < x < 2
Answer Preview: ANSWER a Domain The domain is the set of all possible values of the input variable x for which the f…

, Chapter: 1 -Problem: 29 >> Graph y = sin x and y = [sin x] together. What are the domain and range of [sin x]?
Answer Preview: D R y 1 …

, Chapter: 4 -Problem: 57 >> The geometric mean of two positive numbers a and b is the number ?ab. Show that the value of c in the conclusion of the Mean Value Theorem for ƒ(x) = 1/x on an interval of positive numbers [a, b] is c = ?ab.
Answer Preview: The conclusion …

, Chapter: 4 -Problem: 33 >> Use l’Hôpital’s rule to find the limit. Transcribed Image Text: lim x-0+ 2 In (x² + 2x) In x
Answer Preview: ANSWER To use l Hpital s rule we need to take the de…

, Chapter: 4 -Problem: 38 >> Use l’Hôpital’s rule to find the limit. Transcribed Image Text: lim (lnx - In sin x) x->0+
Answer Preview: ANSWER To apply L Hpital s rule we need to take the derivative of both the numerator and denomi…

, Chapter: 16 -Problem: 3 >> Integrate the given function over the given surface.G(x, y, z) = x2, over the unit sphere x2 + y2 + z2 = 1
Answer Preview: To integrate G x y z x 2 over the unit sphere we need to set up a surface integral Let S be the unit …

, Chapter: 16 -Problem: 13 >> Use the surface integral in Stokes’ Theorem to calculate the flux of the curl of the field F across the surface S in the direction of the outward unit normal n. F = 2zi + 3xj + 5yk S: r(r, ?) = (r cos ?)i + (r sin ?)j + (4 - r2)k, 0 ? r ? 2, 0 ? ? ? 2? Transcribed Image Text: THEOREM 6-Stokes' Theor
Answer Preview: To calculate the flux of the curl …

, Chapter: 11 -Problem: 84 >> Give the eccentricities of conic sections with one focus at the origin of the polar coordinate plane, along with the directrix for that focus. Find a polar equation for each conic section.e = 1/3, r sin ? = -6
Answer Preview: When a conic section has one focus at the origin we have one of the following possible shapes a circ…

, Chapter: 4 -Problem: 105 >> Sketch the graph of a twice-differentiable function y = ƒ(x) with the following properties. Label coordinates where possible. Transcribed Image Text: X x < 2 2 2 < x < 4 4 4 < x < 6 6 x > 6 y 1 4 7 y' < 0, 0, 0, 0, 0, 0, y' Derivatives y² II A A A II V y" y">0 y" y" > 0 y" = 0 y" < 0 y" < 0 y' < 0,
Answer Preview: 7 y 1…

, Chapter: 3 -Problem: 95 >> a. Find the tangent to the curve y = 2 tan(?x/4) at x = 1.b. What is the smallest value the slope of the curve can ever have on the interval -2 < x < 2? Give reasons for your answer.
Answer Preview: a To find the equation of the tangent line to the curve y 2 tan x 4 at x 1 we need to find the slope …

, Chapter: 8 -Problem: 62 >> In a mass-spring-dashpot system like the one in Exercise 61, the mass’s position at time t is Find the average value of y over the interval 0 ? t ? 2?. Exercise 61 A retarding force, symbolized by the dashpot in the accompanying figure, slows the motion of the weighted spring so that the mass’s position at time t is Find the average value of y over the interval 0 ? t ? 2?.
Answer Preview: ANSWER To find the average value of y over the interval 0 t 2 we need to calculate the …

, Chapter: 14 -Problem: 12 >> Use Taylor’s formula to find a quadratic approximation of ex sin y at the origin. Estimate the error in the approximation if |x| ? 0.1 and |y| ? 0.1. Transcribed Image Text: Taylor's Formula for f(x, y) at the Origin ƒ(x, y) = f(0,0) + xfx + yƒy + 1?/17 (x²ƒxx + 2xyƒxy + y²ƒ¸) 2! + 370x³fxxx ³fxxx
Answer Preview: To find a quadratic approximation of ex sin y at the origin using Taylor s formula we ll expand the …

, Chapter: 12 -Problem: 24 >> Let u = i + 2j - k, v = -i + j + k, w = i + k, r = -(?/2)i - ?j + (?/2)k. Which vectors, if any, are (a) Perpendicular? (b) Parallel? Give reasons for your answers.
Answer Preview: a To check whether two vectors are perpendicular we can take their dot product and …

, Chapter: 5 -Problem: 104 >> Find the area of the region between the curve y = 3 - x2 and the line y = -1 by integrating with respect to a. x, b. y.
Answer Preview: a Integrating with respect to x To find the area between the curve y 3 x 2 and the line y 1 we need …

, Chapter: 14 -Problem: 37 >> Find the directions in which ƒ increases and decreases most rapidly at P0 and find the derivative of ƒ in each direction. Also, find the derivative of ƒ at P0 in the direction of the vector v.ƒ(x, y) = cos x cos y, P0(?/4, ?/4), v = 3i + 4j
Answer Preview: To find the directions of the maximum and minimum rate of change at point P0 we need to find the gra…

, Chapter: 11 -Problem: 19 >> Find the lengths of the curves. Transcribed Image Text: x = 3 cos 0, y = 3 sin 0, 0505 ? ? 2
Answer Preview: Since the given curve is in parametric form then its …

, Chapter: 4 -Problem: 9 >> The greatest integer function ƒ(x) = [x], defined for all values of x, assumes a local maximum value of 0 at each point of [0, 1). Could any of these local maximum values also be local minimum values of ƒ? Give reasons for your answer.
Answer Preview: No none of the local maximum values of the greatest integer function x in the interval 0 1 can …

, Chapter: 11 -Problem: 45 >> Sketch the lines and find Cartesian equations for them. Transcribed Image Text: r cos e 7T 4 = ?2
Answer Preview: To sketch the line and find its Cartesian equation we ll start by manipulating the given polar equat…

, Chapter: 9 -Problem: 6 >> Consider the following economic model. Let P be the price of a single item on the market. Let Q be the quantity of the item available on the market. Both P and Q are functions of time. If one considers price and quantity as two interacting species, the following model might be proposed: where a, b, c, and ƒ are positive constants. Justify and discuss the adequacy of the model. a. If a = 1, b = 20,
Answer Preview: a To find the equilibrium points of the system we set both derivatives equal to zero dP dt a p b q p …

, Chapter: 10 -Problem: 24 >> (a) Find the series’ radius and interval of convergence. For what values of x does the series converge  (b) Absolutely,  (c) Conditionally? Transcribed Image Text: ?(In n)xn n=1
Answer Preview: To determine the radius and interval of convergence for the series from to n 1 1 n n x n we can use …

, Chapter: 5 -Problem: 66 >> Use the method of Example 4a or Equation (1) to evaluate the definite integral. Example 4a Compute ?0bx dx and find the area A under y = x over the interval [0, b], b > 0. Transcribed Image Text: To compute the definite integral as the limit of Riemann sums, we calculate lim-0 =1f(c) Ax for parti
Answer Preview: To evaluate the definite integral 0 b x dx we can use Example 4a as follows We have b 0 and we want …

, Chapter: 12 -Problem: 3 >> Find parametric equations for the lines.The line through P(-2, 0, 3) and Q(3, 5, -2)
Answer Preview: To find parametric equations for the line through points P 2 0 3 and Q 3 …

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, Chapter: 1 -Problem: 66 >> Foridentify A, B, C, and D for the sine functions and sketch their graphs. Transcribed Image Text: f(x) = A sin ( ² 7 7 (x ? c ) ) + - C) + D.
Answer Preview: For the function f x A sin 2 B x C D A is the amplitude of the sine function It determines how high …

, Chapter: 1 -Problem: 45 >> Evaluate cos ?/12.
Answer Preview: t…

, Chapter: 4 -Problem: 83 >> Shows the graphs of the first and second derivatives of a function y = ƒ(x). Copy the picture and add to it a sketch of the approximate graph of ƒ, given that the graph passes through the point P. Transcribed Image Text: y P 0 y = f'(x) A y=f"(x) X
Answer Preview: y P I…

, Chapter: 4 -Problem: 27 >> A right triangle whose hypotenuse is ?3 m long is revolved about one of its legs to generate a right circular cone. Find the radius, height, and volume of the cone of greatest volume that can be made this way. Transcribed Image Text: h T V3
Answer Preview: Let s call the leg of the right triangle that we are revolving around x and the other leg y Then we can use the Pythagorean theorem to write x 2 y 2 s…

, Chapter: 4 -Problem: 42 >> Graph the curve. Transcribed Image Text: y = tan -1 1 X
Answer Preview: To draw the graph of the fun…

, Chapter: 15 -Problem: 6 >> Sketch the region of integration and write an equivalent integral with the order of integration reversed. Then evaluate both integrals. Transcribed Image Text: X ff V x dy dx 0x2²
Answer Preview: To sketch the region of integration note that the bounds for x are from 0 to 1 while the bounds for …

, Chapter: 17 -Problem: 19 >> A particle starts at A(-2, 3) and its coordinates change by increments ?x = 5, ?y = -6. Find its new position.
Answer Preview: To find the new position of the particle we need to add the increments x a…

, Chapter: 5 -Problem: 73 >> The marginal cost of printing a poster when x posters have been printed is dollars. Find c(100) - c(1), the cost of printing posters 2–100. Transcribed Image Text: dc 1 dx = 2?x
Answer Preview: We can start by integrating the marginal cost function to get the total cost function c x dc dx dx …

, Chapter: 13 -Problem: 13 >> In r(t) is the position of a particle in space at time t. Find the particle’s velocity and acceleration vectors. Then find the particle’s speed and direction of motion at the given value of t. Write the particle’s velocity at that time as the product of its speed and direction. Transcribed Image Text:
Answer Preview: answer To find the particle s velocity vector we can take the derivative of r t r …

, Chapter: 5 -Problem: 2 >> Write the sums without sigma notation. Then evaluate them. Transcribed Image Text: 3 k=1 k - 1 k
Answer Preview: Expanding the sigma notation w…

, Chapter: 3 -Problem: 57 >> Use a CAS to perform the following step.a. Plot the equation with the implicit plotter of a CAS. Check to see that the given point P satisfies the equation.b. Using implicit differentiation, find a formula for the derivative dy/dx and evaluate it at the given point P.c. Use the slope found in part (b) to find an equation for the tangent line to the curve at P. Then plot the implicit curve and tang
Answer Preview: ANSWER a Using the implicit plotter of a CAS we get the following graph Implicit …

, Chapter: 8 -Problem: 77 >> Evaluate the integrals with  (a) Eq. (4) and  (b) Eq. (5). In each case, check your work by differentiating your answer with respect to x. Transcribed Image Text: S sinh 1 x dx
Answer Preview: We have to evaluate the integral sinh x dx a Using the formula in Eq 4 Let y sinh x Then x sinh y Di…

, Chapter: 15 -Problem: 17 >> Find the center of mass and the moment of inertia about the y-axis of a thin plate bounded by the x-axis, the lines x = ±1, and the parabola y = x2 if ?(x, y) = 7y + 1.
Answer Preview: To find the center of mass and the moment of inertia about the y axis of a thin plate with density f…

, Chapter: 12 -Problem: 22 >> Draw coordinate axes and then sketch u, v, and u * v as vectors at the origin.u = i - j, v = i + j
Answer Preview: Sure I ll describe the coordinate axes and sketch the vectors for you Coordinate axes The x axis is …

, Chapter: 5 -Problem: 79 >> The velocity of a particle moving back and forth on a line is y = ds/dt = 6 sin 2t m/sec for all t. If s = 0 when t = 0, find the value of s when t = ?/2 sec.
Answer Preview: We have the velocity function y ds dt 6 sin 2t m sec Int…

, Chapter: 14 -Problem: 21 >> Draw a branch diagram and write a Chain Rule formula for each derivative. Transcribed Image Text: dw ?s and dw at for w= = h(s, t) g(u), u =
Answer Preview: We have the function w g u where it is a function of the function u The function u is the intermediate variable The functions u have the variables s a…

, Chapter: 11 -Problem: 19 >> What are the standard equations for lines and conic sections in polar coordinates? Give examples.
Answer Preview: In polar coordinates a point is defined by its distance from the origin called the radial coordinate …

, Chapter: 4 -Problem: 13 >> The curve y = tan x crosses the line y = 2x between x = 0 and x = ?/2. Use Newton’s method to find where.
Answer Preview: To use Newton s method we need to start with an initial guess for the soluti…

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, Chapter: 10 -Problem: 79 >> Use a series representation of sin 3x to find values of r and s for which Transcribed Image Text: lim x?0 sin 3x x3 + + S = 0.
Answer Preview: ANSWER We can use the series representation of sin 3…

, Chapter: 10 -Problem: 25 >> Find the Taylor series generated by ƒ at x = a.ƒ(x) = x4 + x2 + 1, a = -2
Answer Preview: To find the Taylor series generated by x x 4 x 2 1 at x a 2 we need to co…

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, Chapter: 12 -Problem: 5 >> Define the dot product (scalar product) of two vectors. Which algebraic laws are satisfied by dot products? Give examples. When is the dot product of two vectors equal to zero?
Answer Preview: ANSWER The dot product also known as the scalar product of two vectors is a scalar quantity …

, Chapter: 1 -Problem: 3 >> What is a piecewise-defined function? Give examples.
Answer Preview: A piecewise defined function is a function that is defined …

, Chapter: 1 -Problem: 42 >> What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing. Transcribed Image Text: = y = x-^
Answer Preview: ANSWER The graph of the function y x has symmetry with respect to the …

, Chapter: 1 -Problem: 34 >> Graph two periods of the function ƒ(x) = 3cot x/2 + 1.
Answer Preview: To graph two periods of the function x 3cot x 2 1 we can start by finding the vertical asymptotes an…

, Chapter: 1 -Problem: 16 >> Find the natural domain and graph the functions.ƒ(x) = 1 - 2x - x2
Answer Preview: ANSWER The given function is a polynomial function of degree two The natural domain of any …

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, Chapter: 4 -Problem: 36 >> What values of a and b make ƒ(x) = x3 + ax2 + bx havea. A local maximum at x = -1 and a local minimum at x = 3?b. A local minimum at x = 4 and a point of inflection at x = 1?
Answer Preview: a For x x 3 ax 2 bx to have a local maximum at x 1 and a local minimum at x 3 the first derivative o…

, Chapter: 4 -Problem: 80 >> Verify the formula differentiation. Transcribed Image Text: dx 1 V?²=sin-¹ (4) + + C
Answer Preview: Proof L H S 1 dx a 2 x 2 Put u x a du 1 a d…

, Chapter: 15 -Problem: 17 >> The counterweight of a flywheel of constant density 1 has the form of the smaller segment cut from a circle of radius a by a chord at a distance b from the center (b < a). Find the mass of the counterweight and its polar moment of inertia about the center of the wheel.
Answer Preview: To find the mass of the counterweight and its polar moment of inertia about the center of the wheel …

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, Chapter: 5 -Problem: 25 >> Let ƒ(x) be a continuous function. Express as a definite integral. Transcribed Image Text: (1) + (a) + - +1(%) 1 lim 11-00
Answer Preview: ANSWER We can rewrite the limit as a Riemann sum and then take the limit as n goes …

, Chapter: 13 -Problem: 6 >> Give the position vectors of particles moving along various curves in the xy-plane. In each case, find the particle’s velocity and acceleration vectors at the stated times and sketch them as vectors on the curve. x2 + y2 = 16 Transcribed Image Text: (4 COS 1)i r(t) = 4 cos (4 sin ½).j j; t = ? and 3
Answer Preview: To find the position vector velocity vector and acceleration vector of a particle moving along the c…

, Chapter: 4 -Problem: 125 >> The standard equation for the position s of a body moving with a constant acceleration a along a coordinate line is where v0 and s0 are the body’s velocity and position at time t = 0. Derive this equation by solving the initial value problem Transcribed Image Text: S 3 = 2/1² + vot + So, (1)
Answer Preview: ANSWER To derive the standard equation for the position of a body we start with the given differe…

, Chapter: 3 -Problem: 106 >> Graph y = -2x sin (x2) for -2 ? x ? 3. Then, on the same screen, graph for h = 1.0, 0.7, and 0.3. Experiment with other values of h. What do you see happening as h ? 0? Explain this behavior. Transcribed Image Text: y = cos ((x + h)²) - cos (x²) h
Answer Preview: ANSWER Now let s consider what happens as h approaches 0 We can see from the graph that a…

, Chapter: 8 -Problem: 57 >> Require the use of various trigonometric identities before you evaluate the integrals. Transcribed Image Text: Isin sin² 0 cos 30 de
Answer Preview: We can start with the identity sin 1 2 1 2 cos 2 So sin cos 3 1 2 1 2 cos 2 co…

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, Chapter: 12 -Problem: 29 >> Describe the given set with a single equation or with a pair of equations.The circle of radius 2 centered at (0, 2, 0) and lying in thea. xy-plane b. yz-plane c. plane y = 2
Answer Preview: a The circle of radius 2 centered at 0 2 0 and lying in the xy plane c…

, Chapter: 5 -Problem: 81 >> Find the areas of the regions enclosed by the curves.4x2 + y = 4 and x4 - y = 1
Answer Preview: To find the …

, Chapter: 14 -Problem: 14 >> Find the maximum and minimum values of 3x - y + 6 subject to the constraint x2 + y2 = 4.
Answer Preview: To find the maximum and minimum values of 3x y 6 subject to the constraint x 2 y 2 4 we can use the …

, Chapter: 11 -Problem: 20 >> The area of the region that lies inside the cardioid curve r = cos ? + 1 and outside the circle r = cos ? is not Why not? What is the area? Give reasons for your answers. Transcribed Image Text: 2T [(cos + 1)² cos²0] de = T
Answer Preview: answer The given integral 1 2 cos 1 2 cos 2 d from 0 to 2 simplifies to 1 2 2 cos 1 d from 0 to 2 1 …

, Chapter: 4 -Problem: 27 >> Use Newton’s method to find the zeros of ƒ(x) = 4x4 - 4x2 using the given starting values.a. x0 = -2 and x0 = -0.8, lying in (-?, -(2/2)b. x0 = -0.5 and x0 = 0.25, lying in (-?21/7, ?21/7)c. x0 = 0.8 and x0 = 2, lying in (?2/2, ?)d. x0 = -?21/7 and x0 = ?21/7
Answer Preview: a Using x0 2 First derivative of f x 4x 4 4x 2 is f x 16x 3 8x The Newton Rap…

, Chapter: 11 -Problem: 55 >> Sketch the parabolas. Include the focus and directrix in each sketch.x2 = -4y
Answer Preview: Sketching the p…

, Chapter: 10 -Problem: 98 >> Assume that each sequence converges and find its limit. Transcribed Image Text: VI. V? + VI, V1 + V1 + V?? V? + V? + V? + VI,... 1 1 V1
Answer Preview: ANSWER For the first sequence notice that as n approaches infinity the deno…

, Chapter: 8 -Problem: 11 >> Verify that the functions are probability density functions for a continuous random variable X over the given interval. Determine the specified probability. Transcribed Image Text: f(x) = xe over [0, ?), P(1 ?X ? 3)
Answer Preview: ANSWER To verify that f x xe is a probability density function …

, Chapter: 11 -Problem: 11 >> Find an equation for the line tangent to the curve at the point defined by the given value of t. Also, find the value of d2y/dx2 at this point.x = t - sin t, y = 1 - cos t, t = ?/3
Answer Preview: To find the equation of the tangent line to the curve at the point defined by t 3 we need to find th…

, Chapter: 5 -Problem: 25 >> Use the Substitution Formula in Theorem 7 to evaluate the integral. Transcribed Image Text: 0 ?/4 (1 + etan) sec²0 de
Answer Preview: ANSWER To evaluate the given definite integral we use the substitution formula i…

, Chapter: 12 -Problem: 3 >> Give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations.y = 0, z = 0
Answer Preview: The given equations are y 0 This is a horizontal plane passing through the x axis z 0 This …

, Chapter: 1 -Problem: 4 >> Which of the functions graphed are one-to-one, and which are not? Transcribed Image Text: y = int x
Answer Preview: The function y int x is not one to one where int x represents the greatest inte…

, Chapter: 1 -Problem: 47 >> Say whether the function is even, odd, or neither. Give reasons for your answer.ƒ(x) = 3
Answer Preview: The function x 3 is an even function because it satisfi…

, Chapter: 1 -Problem: 38 >> What happens if you take B = 2p in the addition formulas? Do the results agree with something you already know?
Answer Preview: If we take B 2p in the addition formulas for elliptic curves the resul…

, Chapter: 1 -Problem: 59 >> A triangle has sides a = 2 and b = 3 and angle C = 60°. Find the length of side c.
Answer Preview: 7…

, Chapter: 4 -Problem: 59 >> a. Find the local extrema of each function on the given interval, and say where they occur. b. Graph the function and its derivative together. Comment on the behavior of ƒ in relation to the signs and values of ƒ?. Transcribed Image Text: f(x) = V3 cos x + sin x, 0?x? 2T
Answer Preview: a Find the local extrema of each function on the given interval and say where they occur As we have …

, Chapter: 4 -Problem: 36 >> a. Find the open intervals on which the function is increasing and decreasing. b. Identify the function’s local and absolute extreme values, if any, saying where they occur. Transcribed Image Text: f(x) = ? 3x² + 1
Answer Preview: ANSWER a To find the intervals where the function is increasing and decreasing we need to find the d…

, Chapter: 4 -Problem: 80 >> a. Prove that ex ? 1 + x if x ? 0. b. Use the result in part (a) to show that Transcribed Image Text: et ? 1 + x + 2.1².
Answer Preview: ANSWER a To prove that e x 1 x if x 0 we will use the Mean Value Theorem for differentiable functions Let f x e x 1 x then f 0 e 0 1 0 0 Now let s consider the interval 0 x where x 0 By the Mean Value Theorem there exists a number c in 0 x such that f c f x f 0 x 0 f c e c 1 Since e c 1 for any c 0 it follows that f c 0 which means that f x f 0 0 or equivalently e x 1 x 0 e x 1 x Therefore we have proved that e x 1 x if x 0 b Using the result in part a we will prove that e x 1 x 1 2 x 2 if x 0 Let s consider the function …

, Chapter: 17 -Problem: 32 >> Describe the regions defined by the inequalities and pairs of inequalitie.(x - 1)2 + y2 ? 4
Answer Preview: The given inequality is x 1 y 4 This inequality represents a circular regio…

, Chapter: 5 -Problem: 13 >> Find the total area of the region between the graph of ƒ and the x-axis.ƒ(x) = 5 - 5x2/3, -1 ? x ? 8
Answer Preview: To find the total area of the region between the graph of x and the x axis we need to integrate the …

, Chapter: 12 -Problem: 51 >> Use a calculator to find the acute angles between the planes to the nearest hundredth of a radian.2x + 2y - z = 3, x + 2y + z = 2
Answer Preview: To find the acute angle between two planes we need to find the cosine of the angle between their nor…

, Chapter: 5 -Problem: 5 >> Write the sums without sigma notation. Then evaluate them. Transcribed Image Text: 3 k=1 (-1)*+¹ sin TT k
Answer Preview: ANSWER The given series is k 1 to 3 …

, Chapter: 3 -Problem: 44 >> Find equations for the tangent and normal to the cissoid of Diocles y2(2 - x) = x3 at (1, 1). Transcribed Image Text: y 1 0 3 y²(2-x) = x³ 1 (1, 1)
Answer Preview: To find the equations for the tangent and normal to the cissoid of Diocles y 2 2 x x 3 at 1 1 we fir…

, Chapter: 8 -Problem: 49 >> A manufacturer of generator shafts finds that it needs to add additional weight to its shafts in order to achieve proper static and dynamic balance. Based on experimental tests, the average weight it needs to add is ? = 35 gm with ? = 9 gm. Assuming a normal distribution, from 1000 randomly selected shafts, how many would be expected to need an added weight in excess of 40 gm?
Answer Preview: answer The weight that needs to be added to a shaft has a normal distribution with …

, Chapter: 15 -Problem: 1 >> Evaluate the iterated integral. Transcribed Image Text: 2 4 SS /0 2xy dy dx
Answer Preview: Given iterated integral is 0 1 x 1 x 2y 2 dy …

, Chapter: 12 -Problem: 31 >> Find the vectors whose lengths and directions are given. Try to do the calculations without writing. Transcribed Image Text: Length a. 2 b. ?3 1 2 d. 7 C. Direction i -k 3 + k ²7/? + k 7
Answer Preview: Here are the solutions to the given problem a The vector whose length is 2 and direction is in the i …

, Chapter: 5 -Problem: 78 >> Find ƒ(4) if ?0x ƒ(t) dt = x cos ?x.
Answer Preview: We are given that 0x t dt x cos x To find 4 we need to evaluate the defin…

, Chapter: 14 -Problem: 19 >> Find the minimum distance from the surface x2 - y2 - z2 = 1 to the origin.
Answer Preview: To find the minimum distance from the surface x 2 y 2 z 2 1 to the origin we need to find the distan…

, Chapter: 11 -Problem: 20 >> Graph the sets of points whose polar coordinates satisfy the equations and inequalitie.? = ?/2, r ? 0
Answer Preview: The equation 2 represents the line passing through the pole 0 0 …

, Chapter: 4 -Problem: 3 >> Does ƒ(x) = (7 + x)(11 - 3x)1/3 have an absolute minimum value? An absolute maximum? If so, find them or give reasons why they fail to exist. List all critical points of ƒ.
Answer Preview: No minimum a…

, Chapter: 11 -Problem: 60 >> Replace the Cartesian equations with equivalent polar equations.xy = 2
Answer Preview: To convert the Cartesian equation xy 2 to a polar equation we can use the conversion formul…

, Chapter: 10 -Problem: 103 >> A triple of positive integers a, b, and c is called a Pythagorean triple if a2 + b2 = c2. Let a be an odd positive integer and let be, respectively, the integer floor and ceiling for a2/2. a. Show that a2 + b2 = c2.  b. By direct calculation, or by appealing to the accompanying figure, find Transcribed Image Text:
Answer Preview: a b a 2 2 c a 2 2 b 2 a 4 4 an…

, Chapter: 3 -Problem: 32 >> Verify that the given point is on the curve and find the lines that are (a) Tangent  (b) Normal to the curve at the given point.x2 + y2 = 25, (3, -4)
Answer Preview: To verify that the point 3 4 is on the curve we substitute x 3 and y 4 into the equation x 2 y 2 25 …

, Chapter: 10 -Problem: 28 >> What is Taylor’s formula? What does it say about the errors involved in using Taylor polynomials to approximate functions? In particular, what does Taylor’s formula say about the error in a linearization? A quadratic approximation?
Answer Preview: Taylor s formula also known as Taylor s theorem is a mathematical result that provides a way to appr…

, Chapter: 5 -Problem: 58 >> Graph the function and find its average value over the given interval.ƒ(x) = 3x2 - 3 on [0, 1]
Answer Preview: To graph the function x 3x 2 3 on 0 1 we can start by finding its crit…

, Chapter: 12 -Problem: 7 >> Give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations.x2 + z2 = 4, y = 0
Answer Preview: The given equations describe a circular cylinder of …

, Chapter: 1 -Problem: 36 >> Suppose that in any given year the number of cases of a disease is reduced by 20%. If there are 10,000 cases today, how many years will it takea. To reduce the number of cases to 1000?b. To eliminate the disease; that is, to reduce the number of cases to less than 1?
Answer Preview: a Let P be the number of cases of the disease after t years The…

, Chapter: 1 -Problem: 15 >> What is the period of each function?cos ?x
Answer Preview: Perio…

, Chapter: 4 -Problem: 67 >> L’Hôpital’s Rule does not help with the limit. Try it—you just keep on cycling. Find the limits some other way. Transcribed Image Text: lim ?9x + 1 ?x + 1
Answer Preview: To evaluate the limit of 9x 1 x 1 as x approaches infinity we can use the fact that for …

, Chapter: 4 -Problem: 35 >> Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. Transcribed Image Text: y = x2/3 5 NIU X
Answer Preview: Infl 1 2 3 4 2 …

, Chapter: 4 -Problem: 77 >> Only one of these calculations is correct. Which one? Why are the others wrong? Give reasons for your answers. a. b. c. d. Transcribed Image Text: lim x ln x = 0·(-?) = 0 x->0+
Answer Preview: a this option is wrong because if 0 is multiplie…

, Chapter: 10 -Problem: 30 >> Use any method to determine if the series converges or diverges. Give reasons for your answer. Transcribed Image Text: ? n=1 ? n n
Answer Preview: The convergence of the given series 1 n 1 n 2 n n 1 to can be dete…

, Chapter: 16 -Problem: 10 >> Integrate the given function over the given surface. Integrate G(x, y, z) = y + z over the surface of the wedge in the first octant bounded by the coordinate planes and the planes x = 2 and y + z = 1.
Answer Preview: To integrate the function G x y z y z over the surface of the wedge in the first octant bounded by t…

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, Chapter: 11 -Problem: 74 >> Find the volume of the solid generated by revolving the region enclosed by the ellipse 9x2 + 4y2 = 36 about the (a) x-axis, (b) y-axis.
Answer Preview: We can use the method of cylindrical shells to find the volume of the solid generated by revolving the region enclosed by the ellipse 9x 2 4y 2 36 abo…

, Chapter: 4 -Problem: 111 >> Suppose the derivative of the function y = ƒ(x) is At what points, if any, does the graph of ƒ have a local minimum, local maximum, or point of inflection? Transcribed Image Text: y' = (x - 1)²(x - 2).
Answer Preview: ANSWER To find the local extrema and points of inflection of the function x we need to analyze the s…

, Chapter: 3 -Problem: 51 >> The radius of a circle is increased from 2.00 to 2.02 m.a. Estimate the resulting change in area.b. Express the estimate as a percentage of the circle’s original area.
Answer Preview: a To estimate the resulting change in area when the radius of a circle is increased from 2 00 to 2 0…

, Chapter: 8 -Problem: 60 >> Consider the region bounded by the graphs of y = tan-1 x, y = 0, and x = 1.a. Find the area of the region.b. Find the volume of the solid formed by revolving this region about the y-axis.
Answer Preview: ANSWER a To find the area of the region we need to integrate the function y tan 1 x from x 0 to x 1 …

, Chapter: 15 -Problem: 14 >> Convert the integral to an equivalent integral in cylindrical coordinates and evaluate the result. Transcribed Image Text: V1-y² [[[[ 0 0 (x² + y2) dz dx dy ²
Answer Preview: The given integral is R x 2 y 2 dA where R is the region in the xy plane bou…

, Chapter: 12 -Problem: 24 >> For what value or values of a will the vectors u = 2i + 4j - 5k and v = -4i - 8j + ak be parallel?
Answer Preview: If two non zero vectors are parallel then they must be …

, Chapter: 14 -Problem: 20 >> Find the partial derivative of the function with respect to each variable. Transcribed Image Text: f(x, y) = = -1/1 In (x² + y²) + tan -1 y X
Answer Preview: To find the partial derivative of the function f x y with respect to x we differentiate f x …

, Chapter: 11 -Problem: 16 >> What is a hyperbola? What are the Cartesian equations for hyperbolas centered at the origin with foci on one of the coordinate axes? How can you find the foci, vertices, and directrices of such an ellipse from its equation?
Answer Preview: A hyperbola is a conic section curve that consists of two separate curves called branches that are m…

, Chapter: 4 -Problem: 14 >> The Second Derivative Test for Local Maxima and Minima says: a. ƒ has a local maximum value at x = c if ƒ?(c) = 0 and ƒ?(c) b. ƒ has a local minimum value at x = c if ƒ?(c) = 0 and ƒ?(c) > 0. To prove statement (a), let P = (1/2) |ƒ?(c)|. Then use the fact that to conclude that for some ? > 0, Thus, ƒ?(c + h) is positive for -?
Answer Preview: ANSWER To prove statement a we need to show that if c 0 and c 0 then has a local maximu…

, Chapter: 11 -Problem: 59 >> Replace the Cartesian equations with equivalent polar equations. Transcribed Image Text: + y 4 = 1
Answer Preview: To replace the given Cartesian equation with an equivalent …

, Chapter: 10 -Problem: 88 >> Which of the sequences {an} converge, and which diverge? Find the limit of each convergent sequence. Transcribed Image Text: 1 an ??n² ? 1 = ???n² + n - -
Answer Preview: ANSWER To determine if the sequence an converges or diverges we …

, Chapter: 3 -Problem: 23 >> Use implicit differentiation to find dy/dx and then d2y/dx2.y2 = ex2 + 2x
Answer Preview: To find dy dx and d2y dx2 using implicit differentiation we need to differentiate bo…

, Chapter: 10 -Problem: 29 >> Which of the series converge, and which diverge? Give reasons for your answers. Transcribed Image Text: ? n=1 1 (In 2)"
Answer Preview: ANSWER To determine the convergence or divergence of t…

, Chapter: 5 -Problem: 44 >> Use the rules in Table 5.6 and Equations (2)–(4) to evaluate the integral. Transcribed Image Text: 0 ?2 (t-?2) d dt
Answer Preview: Use the rules in Table 5 6 and Equations 2 4 to evaluate the …

, Chapter: 12 -Problem: 9 >> What geometric or physical interpretations do cross products have? Give examples.
Answer Preview: A mathematical procedure called the cross product produces a new vector that is orthogonal to both o…

, Chapter: 1 -Problem: 30 >> Graph y = sin x and y = [sin x] together. What are the domain and range of [sin x]?
Answer Preview: To graph y sin x and y sin x we need to graph them separately and then combine the …

, Chapter: 1 -Problem: 19 >> What is the period of each function? Transcribed Image Text: COS X TT 2
Answer Preview: Perio…

, Chapter: 4 -Problem: 71 >> Discuss the extreme-value behavior of the function ƒ(x) = x sin (1/x), x ? 0. How many critical points does this function have? Where are they located on the x-axis? Does ƒ have an absolute minimum? An absolute maximum?
Answer Preview: The function x x sin 1 x is an example of a pathological function with interesting extreme value beh…

, Chapter: 4 -Problem: 20 >> a. Find the open intervals on which the function is increasing and decreasing.b. Identify the function’s local and absolute extreme values, if any, saying where they occur.g(t) = -3t2 + 9t + 5
Answer Preview: ANSWER a To find the open intervals on which the function is increasing and decreasing we need to ta…

, Chapter: 4 -Problem: 82 >> Shows the graphs of the first and second derivatives of a function y = ƒ(x). Copy the picture and add to it a sketch of the approximate graph of ƒ, given that the graph passes through the point P. Transcribed Image Text: Po y = f'(x) y = f"(x) X
Answer Preview: We are given the graphs of f x

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