Fall 2016, MA 252, Calculus II, Final Exam Preview Solutions

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1 Fall 6, MA 5, Calculus II, Final Exam Preview Solutions I will put the following formulas on the front of the final exam, to speed up certain problems. You do not need to put them on your index card, and you should learn to recognize them and use them whenever possible, to save time. sec 3 (x) dx sec(x) tan(x) + ln sec(x) + tan(x) csc(x) dx ln csc(x) + cot(x) xe x dx xe x x x tan (x) dx x + tan (x) x a x dx x ( a x + a x ) sin a x a dx x x a a x ln + x a x + a dx x x + a + a x ln + x + a In addition to the following problems, I would recommend that you look at the practice tests for the three midterms. However, I will not give you the very hardest integrals on the final.. Let R be the region in the first quadrant which is bounded by y x, the x-axis and x. (a) Sketch the region R and find its area. (b) Sketch the volume of revolution V generated by rotating R around the x-axis and find the volume. (c) Sketch the volume of revolution V generated by rotating R around the y-axis and find the volume. (d) Set up an integral, but do not solve it, for the volume of revolution generated by rotating R around the line x. (a) Here s a graph of the region

2 MA 5, Calculus II, Final Exam Preview, Fall 6 Now we find the area. x dx x3 3 (b) Here s a graph of the volume of rotation. 3 Now we calculuate the volume using discs. V πr dx π π x5 5 π 5 π(x ) dx x dx (c) Here s a graph of the volume of rotation:

3 MA 5, Calculus II, Final Exam Preview, Fall 6 3 Now we calcluate the volume using shells. πrh dx (d) Here s a sketch of the volume πx x dx π x 3 dx π x π We use shells again πrh dx π( x)x dx. Find the following integral sec (x) tan (x) dx

4 MA 5, Calculus II, Final Exam Preview, Fall 6 u tan(x) du sec (x) dx sec (x) tan (x) dx u du u3 3 3 tan3 (x) 3. Find the following integral x 3 5 x dx. u x du 8x 3 dx x 3 5 x dx u /5 du u6/5 5 8 (x ) 6/5. Find the following integral xe 3x dx. Integration by parts f g f g f g f x g e 3x f xe 3x dx x 3 e3x 3 e3x x 3 e3x 9 e3x g 3 e3x

5 6. Find the following integral x dx MA 5, Calculus II, Final Exam Preview, Fall Find the following integral Dividing gives x x 3x + dx Thus we find R 3x x 3x + x + x + (x 3x + ) 3x + 3x x dx. We use partial fractions: 3x + 3x (x )(x ) A x + B x 3x A(x ) + B(x ) Plugging in x gives B so B. Plugging in x gives A. Thus we find + x dx x + ln x ln x x (Hint: you will need to know the integrals sec(θ) dθ and sec 3 (θ) dθ.) Note: the right way to do this problem on the final would be to use one of the integral formulas I ve put on the front. But perhaps you want to practice trig substitution as well. I ll show here how to practice. Let x sec(θ). Then dx sec(θ) tan(θ) and x sec (θ) tan(θ). Thus the integral becomes tan(θ) sec(θ) tan(θ) dθ tan (θ) sec(θ) dθ The methods of 7. don t seem to help here (that is, neither u sec(θ) nor u tan(θ) seem to help). Thus, we will need to use the identity tan (θ) + sec (θ) and see if this helps. It does. (sec (θ) ) sec(θ) dθ sec 3 (θ) sec(θ) dθ Now we look up these integrals and get ( sec(θ) tan(θ) + ) ln sec(θ) + tan(θ) ln sec(θ) + tan(θ)

6 MA 5, Calculus II, Final Exam Preview, Fall 6 6 which simplifies as ( ) sec(θ) tan(θ) ln sec(θ) + tan(θ) Now we need to get back to x s. Our original substitution was x sec(θ), which gives that sec(θ) x Since x sec(θ) we have sec(θ) x whence x x θ x whence tan(θ). So, our final solution is ( x x ) x ln + x or x x ln x + x 7. Determine whether or not the following integral exists. If it exists, find it s value. Be sure to use lim correctly x + x dx u + x x u du x dx x u x dx + x u du ln(u) lim u ( ) lim ln(u) ln() u (DNE ) DNE Since lim ln(u) does not exist, the integral does not exist. u

7 MA 5, Calculus II, Final Exam Preview, Fall Find the arc-length of the curve y x from x to x. A.L. + (f (x)) dx + (x) dx u x x u du dx x u A.L. + u du The right way to finish this problem is to use one of the integrals on the front. But, if you want to practice your trig substitution, then read on. u tan(θ) + u du du sec (θ) dθ + tan (θ) sec (θ) dθ sec 3 (θ) dθ (sec(θ) tan(θ) + ln sec(θ) + tan(θ) ) (see Lecture notes, 7., Example 5) [ u ] + u + ln + u + u ( 5 + ln( + 5)) By the way, no, I won t put a problem this hard on the final exam! 9. Find the median of the following exponential probability density function: f(t) e t/ for t This means solving for b so that b for all values of t <. Thus, our integral becomes b e t/ dt f(x). Our function equals

8 MA 5, Calculus II, Final Exam Preview, Fall 6 8 Now we just take the anti-derivative and solve for b e t/ b (e b/ e ) e b/ e b/ / b ln(/). I borrow $3,56 from the bank, and agree to pay it back over 3 months. Each month, I will be charged.5%, compound interest, and I will pay $. Thus, at the end of one month I will owe At the end of two months I will owe 356( +.5) 356( +.5) ( +.5) At the end of three months I will owe 356( +.5) ( +.5) ( +.5) At the end of N months I will owe N 356( +.5) N ( +.5) n n (a) Calculate how much I will owe after month. (b) Calculate how much I will owe after months. (c) Calculate how much I will owe after 5 months. (d) Calculate how much I will owe after months. (a) (b) 356( +.5) ( +.5) ( +.5) (c) 9 356(+.5) 5 (+.5) n 356(.5) 5 (.5) n

9 MA 5, Calculus II, Final Exam Preview, Fall 6 9 (d) (+.5) (+.5) n 356(.5) (.5) n. (a) Can you apply the Divergence test to the following? If so, then apply it. Or does it not apply? n sin(/n) n (b) Can you apply the Divergence test to the following? If so, then apply it. Or does it not apply? sin(/n) n (a) As you try to size this one up, you should be a little confused, what the heck does lim n sin(/n) look like? n lim n sin(/n) for L Hospital s Rule n sin(/n) lim n /n L. H. cos(/n)( /n ) lim n /n lim cos(/n) cos() n Therefore this series diverges by the Divergence Test. By the way, now that you know Taylor series, you can find this limit without L Hospital s rule. Just use sin(x) x for x. (b) lim sin(/n) sin(/ ) n You cannot use the Divergence Test here. The Divergence test says If lim then.... We cannot apply that because we do not have If lim. So, we can draw no conclusions.. Apply the Ratio Test to each of the following: (a) n (n ) n!

10 MA 5, Calculus II, Final Exam Preview, Fall 6 (b) Note that the top is supposed to represent the product of odd numbers, starting at and going up to n. If you want, you can write this as a double factorial: (n )!!. (a) lim n ((n+) ) (n+)! (n ) n! (7n n + ) 5 n n (n)! lim (n + ) n (n ) n! (n + )! lim (n )(n + ) 3... n n (n ) 3... n(n + ) lim n + n n + Since L, and since >, the series diverges. 3. Find the radius and open interval of convergence for each of the following series: (a) (b) n ( ) n n n (x + 3) n n (x 8)n n n (a) ( ) n+ (n+) (x + 3) n+ lim n+ n ( ) n n (x + 3) n n lim n lim ( ) n ( ) n+ (n + ) ( ) n n x + 3 L x + 3 x + 3 < x + 3 < R C 3 (n + ) n IOC ( 3, 3 + ) ( 7, ) n (x + 3)n+ n+ (x + 3) n (x + 3)

11 MA 5, Calculus II, Final Exam Preview, Fall 6 (b) lim n n+ n+ n n (x 8)n+ (x 8)n x 8 < lim n lim n n+ n n n + (x 8) L x 8 x 8 < / x 8 < x < 8 (x 8)n+ (x 8) n (x 8) n n + R 8 C ( IOC 8, + ) ( 7/8, /8) 8. Find the Taylor polynomial of e x at a, write your answer in notation. We take the derivative of f(x) e x a few times, plug in x a few times, and then divide by the appropriate factorial to find the coefficient c n f (n) (). n! Thus we have n f (n) (x) f (n) () c n f (n) () n! e x e c e x e c e e x e c e/! 3 e x e c 3 e/3! e x e c e/! e x e + e(x ) + e! (x ) + e 3! (x )3 + e! (x ) +... n (x )n e n! 5. Do ONE OF THE FOLLOWING PARTS (a) e x x has a solution between and. Use a degree Macluarin polynomial to find an algebraic approximation of this solution.

12 MA 5, Calculus II, Final Exam Preview, Fall 6 (b) Use degree 3 Maclaurin polynomial to algebraically find the following limit: x 5 sin(x 5 ) lim x x 5 (c) Use a Maclaurin polynomial to find an algebraic approximation of sin(x) x Use five nonzero terms in your answer. dx 6. e x + x + x! e x x + x + x! x x + x + x + x + x ± 6 ()() () ± 8 ± 7. sin(x) x x3 3! sin(x 5 ) x 5 (x5 ) 3 x 5 x5 3! x 5 sin(x 5 ) lim x x 5 lim 3! x x5 lim x lim x 3! 3! ( 3! x5 x 5 x 5 x5 3! x 5 )

13 MA 5, Calculus II, Final Exam Preview, Fall sin(x) x x3 3! + x5 5!... sin(x) x dx x x3 3! + x5 5!... dx x x 3! + x 5!... dx ) (x x3 3 3! + x5 5 5! ! + 5 5! 7 7! + 9 9! 9. (a) Find the equation of the tangent line at θ π/3, to the following polar function (decimal answers are ok here): r + cos(θ) (b) Graph the function, and it s tangent line, in the same graph. (a) y m(x x ) + y x cos(π/3)( + cos(π/3)) y sin(π/3)( + cos(π/3)) 3.73 m dy/dθ dx/dθ cos(θ)( + cos(θ)) sin (θ) sin(θ)( + cos(θ)) cos(θ) sin(θ) / /( 3) y.9(x ) +.73 θπ/3

14 MA 5, Calculus II, Final Exam Preview, Fall 6 (b) XT cos(t )( + cos(t )) Y T sin(t )( + cos(t )) XT T Y T.9(T ) Find the area of the shaded region r + cos(θ)

15 MA 5, Calculus II, Final Exam Preview, Fall 6 5 A π π π r dθ ( + cos(θ) dθ + cos(θ) + cos (θ) dθ 3 + sin(θ) + 8 sin(θ) π 3 π. Match the following equations to the graphs below. (a) x t + t, y t t (b) x e t + t, y e t t (c) x t, y t (d) x. + cos(t), y. tan(t) + sin(t) (e) x t sin(t), y cos(t) (a) V. Both x and y values are almost always positive (just barely negative), and increase without bound. (b) IV. Both x and y are always positive and increase without bound. (c) III. Both x and y are linear. (d) I. x is bounded between two numbers. y is unbounded, both negative and positive. (e) II. x is unbounded, growing with t. y is bounded, between and 3.

16 MA 5, Calculus II, Final Exam Preview, Fall 6 6 I II III IV V

17 MA 5, Calculus II, Final Exam Preview, Fall 6 7. Find the equation of the tangent line for the parametric curve below at the point t 5π/6 (decimals ok) x cos(t) + cos(t) y sin(t) + sin(t) y m(x x ) + y x y m dy/dt dx/dt cos(t) + cos(t) sin(t) sin(t) 3/ 3 /.9 t5π/6 y.9(x +.366).366 The graph wasn t asked for in this problem, but here it is if you want to double check your work:

18 MA 5, Calculus II, Final Exam Preview, Fall Find the exact length of the curve AL x e t + e t, y 5 t, t 3 (dx/dt) + (dy/dt) dt (et e t ) + dt e t + e t + dt e t + + e t dt (et + e t ) dt e t + e t dt e t e t 3 e 3 e 3 ( ) e 3 e 3