Math 3 Calculus II University of Colorado Spring Final eam review problems: ANSWER KEY. Find f (, ) for f(, y) = esin( y) ( + y ) 3/.. Consider the solid region W situated above the region apple apple, apple y apple, and bounded above by the surface z = e. An equivalent description for the region apple apple, apple y apple is: apple y apple, y apple apple. (a) Write an integral that evaluates the area of each cross-section of W with vertical planes y = a, where a is a constant in [, ]. Can you evaluate this integral (as an epression depending on a)? In each cross-sectional plane y = a, the interval for is given by a apple apple, so the area of the cross-sections is Z a f(, a) d = Z a e d, which can t be evaluated directly. (b) Write an integral that evaluates the area of each cross-section of W with vertical planes = b, where b is a constant in [, ]. Can you evaluate this integral (as an epression depending on b)? In each cross-sectional plane = b, the interval for y is given by apple y apple b, so the area of the cross-sections is Z b f(b, y) dy = Z b e b dy = be b. (c) Write a convenient double integral that evaluates the volume of the region W, and evaluate it. Z Z R zda = Z Z e dy d = Z Z e y d dy The previous remarks indicate that the first option is the only convenient one, so the volume of the solid region can be calculated as: Z Z Z e dy d = ye y= y= d = Z e d = e = e4 (d) Let f(, y) = p + y. (a) Find f (, y). / p + y. (b) Let g() = f (, ). Does lim! g() eist? If so, evaluate it. If not, eplain why not. Hint: Note that f(, ) = p =. No. 3. Evaluate: The limit does not eist because lim g() = lim!! = p = lim!. 8 < = if <, ( ) : = if >.
(a) (b) 87 4 8 4e3 + 9e 6 Z Z y y Z Z e e y ( + y) d dy. + e ln d dy. 4. (a) Set up a double integral that corresponds to the mass of the semidisk D = {(, y): apple y apple p 4 }, if the density at each of its points is given by the function (, y) = y. (b) Find the mass of the above semidisk. 64 5 Z Z p 4 y dy d 5. The power series X C n n diverges at = 7 and converges at = 3. At = 9, the series is (a) Conditionally convergent (b) Absolutely convergent (c) Divergent (d) Cannot be determined. 6. The power series X C n ( 5) n converges at = 5 and diverges at =. At =, the series is (a) Conditionally convergent (b) Absolutely convergent (c) Divergent (d) Cannot be determined. 7. The power series X C n n diverges at = 7 and converges at = 3. At = 4, the series is (a) Conditionally convergent (b) Absolutely convergent (c) Divergent (d) Cannot be determined. 8. In order to determine if the series X k= p k + k + converges or diverges, the limit comparison test can be used. Decide which series provides the best comparison.
(a) (b) (c) X k= X k= X k= k k 3/ k 9. Decide whether the following statements are true or false. Give a brief justification for your answer. (a) If f is continuous for all and TRUE Z f()d converges, then so does Z a f()d for all positive a. Z (b) if f() is continuous and positive for > and if lim f() =, then f()d converges.! FALSE (c) If (d) If Z Z f()d and R g()d both converge then f()d and Z g()d both diverge then Z Z (f() + g())d converges. TRUE (f() + g())d diverges. FALSE. For this problem, state which of the integration techniques you would use to evaluate the integral, but DO NOT evaluate the integrals. If your answer is substitution, also list u and du; if your answer is integration by parts, also list u, dv, du and v; if your answer is partial fractions, set up the partial fraction decomposition, but do not solve for the numerators; if your answer is trigonometric substitution, write which substitution you would use. Z (a) tan d Use substitution w = cos, so that dw = sin Z d (b) 9 Factor out the denominator as 9 = ( 3)( + 3), and use partial fractions to rewrite the integrand as 9 = A 3 + B + 3. Z (c) e cos d Start by using an integration by parts, with u = e and dv = cos d, so that du = e d and v = sin. (Then use another integration by parts with U = e and dv = sin d, and solve the resulting equation for the integral.) (d) (e) (f) (g) Z p 9 d Start by using a trig substitution = 3 sin t, so that d = 3 cos tdt. Z sin(ln ) d Substitute w = ln, so that dw = d. Z 3/ ln d Use integration by parts, with u = ln and dv = 3/ d, so that du = and v = 5 5/. Z p + 4 d Trig substitution = tan t, so that d = sec t. (h) Z e + e + d The numerator is the derivative of the denominator. Use substitution w = e +, so that dw = (e + )d. 3
. If X a n ( ) n = ( ) n= ( ) + ( )3 4 ( ) 4 + 8 ( )5 6 then the correct formula for a n is: Oops, in the original version of this revew sheet, all of the possible answers below had a factor of ( ) n. Please ignore those. (a) a n = ( )n n (b) a n = ( )n+ n (c) a n = ( )n n (d) a n = ( )n+ n. True or False? If lim n! a n = then (a) True (b) False 3. True or False? If lim n! a n 6= then (a) True (b) False X a n converges. n= X a n diverges. n= 4. True/False: If X a n is convergent, then the power series P a n n has convergence radius at least R =. True X ( ) n n+ 5. If one uses the Taylor polynomial P 3 () of degree n = 3 to approimate sin = at (n + )! =., would one get an overestimate or an underestimate? underestimate 6. Suppose that is positive but very small. Arrange the following in increasing order:,, sin, ln( + ), cos(), e, p. cos < p < ln( + ) < sin < e 7. (a) Find the Taylor series about for f() = e. e = (b) Is this function even or odd? Justify your answer. even (c) Find f (3) () and f (6) (). f (3) () =, f (6) () = 6!! X n= n+ 8. Suppose that ibuprofen is taken in mg doses every si hours, and that all mg are delivered to the patient s body immediately when the pill is taken. After si hours,.5% of the ibuprofen remains. Find epressions for the amount of ibuprofen in the patient immediately before and after the n th pill taken. Include work; without work, you may receive no credit. Just before the nth dose: n! 4
nx (.5 +.5 +.5 3 + +.5 n ) =.5 k =.5n.5 Just after the nth dose: +.5n.5 9. Find an equation for a sphere if one of its diameters has endpoints (,, 4) and (4, 3, ). ( 3) + (y ) + (z 7) =. A cube is located such that its top four corners have the coordinates (,, ), (, 3, ), (4,, ), and (4, 3, ). Give the coordinate of the center of the cube. (.5,.5,.5). Find the equation of the largest sphere with center (5, 4, 9) contained in the first octant. ( 5) + (y 4) + (z 9) = 6. Evaluate the double integral (using the most convenient method): k= (a) (b) Z Z e e Z y Z ln d dy e y ln dy d + y4 4 3. The following sum of double integrals describes the mass of a thin plate R in the y-plane, of density (, y) = + y: mass = Z Z +8 4 (, y) dy d + Z 4 Z +8 (, y) dy d (a) Describe the thin plate (shape, intersections with the coordinate aes). It s an isosceles triangle with base along the ais from 4 to 4, and ape on the y ais at (, 4). (b) Write an epression for the mass of the plate as only one double integral. Z 4 Z (8 y)/ (y 8)/ ( + y) d dy (c) Calculate the area and the mass of the plate. 56/3 4. If one uses the Taylor polynomial P 5 () of degree n = 5 to approimate e = would one get an overestimate or an underestimate? underestimate X n n! at =., 5. A slope field for the di erential equation y = y e is shown. Sketch the graphs of the solutions that satisfy the given initial conditions. Make sure to label each sketched graph. (a) y() = RED (b) y() = GREEN (c) y() = BLUE 5
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6. For each di erential equation, find the corresponding slope field. (Not all slope fields will be used.) A Equation Slope Field B dy d = B y = y F y = sin() D C dy d = y A D E F 7. Let f(, y) = e y+y. (a) Find f (, y). e y+y (y + y ) (b) Find f y (, y). e y+y ( + y) (c) Find @ @ f y(, y). e y+y ( 3 y + 5 y + y 3 + + y) (d) Find @ @y f (, y). e y+y ( 3 y + 5 y + y 3 + + y)what s curious about the relation of this answer to that of part (c) above? They re equal. 8. Find the equation of the sphere that passes through the origin and whose center is (,, 3). ( ) + (y ) + (z 3) = 4. 7
9. You want to estimate sin using the first 3 nonzero terms in the Taylor series. What formula for the error bound would you use to get the best estimate for the error? E 6 () apple 7 7!. 3. Find the arc length of the curve y = 3/ from (, ) to (, p ). 8 7 " 3 4 3/ # 3/ 3 4 3. Find the arc length of the curve y = ( 6 + 8)/(6 ) from = to = 3. 3. Let f(, y) = 5 3 y + 5y 4. Show that @ @ f (, y) + @ @y f y(, y) =. 595 44 @ @ f (, y) = 3 6y ; @ @y f y(, y) = 6y 3. 33. Additional problems from your tet: (a) Section 8. (Areas and Volumes) Eercises 3,, 5 8
(b) Section 8. (Volumes by Revolution & Cross Sections) Eercises 9, 3, 35 9
(c) Section 8.3 (Arc Length and Parametric Curves) Eercises, 5, 7, 9
(d) Section.4 (Separation of Variables) Eercises 3, 9, 3,, 4
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