Spring 5, 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
MA 5, Calculus II, Final Exam Preview, Spring 5 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 4 dx (c) Here s a graph of the volume of rotation:
MA 5, Calculus II, Final Exam Preview, Spring 5 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 x4 4 π We use shells again πrh dx π( x)x dx. Find the following integral sec (x) tan (x) dx
MA 5, Calculus II, Final Exam Preview, Spring 5 4 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 4 dx. u x 4 du 8x 3 dx x 3 5 x 4 dx u /5 du 8 8 5 6 u6/5 5 48 (x4 ) 6/5 4. 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
MA 5, Calculus II, Final Exam Preview, Spring 5 5 5. Find the following integral Thus we find Dividing gives x x 3x + dx 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 4 A. Thus we find + 4 x dx x + 4 ln x ln x x 6. Find the following integral x 4 dx (Hint: you will need to know the integrals sec(θ) dθ and sec 3 (θ) dθ. We do trig substitution. Let x sec(θ). Then dx sec(θ) tan(θ) and x 4 4 sec (θ) 4 tan(θ). Thus the integral becomes tan(θ) sec(θ) tan(θ) dθ 4 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. 4 (sec (θ) ) sec(θ) dθ 4 sec 3 (θ) sec(θ) dθ Now we look up these integrals and get ( 4 sec(θ) tan(θ) + ) ln sec(θ) + tan(θ) ln sec(θ) + tan(θ) which simplifies as ( ) sec(θ) tan(θ) ln sec(θ) + tan(θ)
MA 5, Calculus II, Final Exam Preview, Spring 5 6 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 4 θ x 4 whence tan(θ). So, our final solution is ( x x 4 ) x ln + x 4 or x x 4 ln x + x 4 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 8. Find the arc-length of the curve y x from x to x.
MA 5, Calculus II, Final Exam Preview, Spring 5 7 A.L. Trig-substitution + (f (x)) dx + (x) dx u x x u du dx x u A.L. + u du 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) 4 [ u ] + u 4 + 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
MA 5, Calculus II, Final Exam Preview, Spring 5 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 $3,456 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 3456( +.5) 3456( +.5) ( +.5) At the end of three months I will owe 3456( +.5) ( +.5) ( +.5) At the end of N months I will owe N 3456( +.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) 3456( +.5) 373.8 3456( +.5) ( +.5) 688.65 (c) 49 3456(+.5) 5 (+.5) n 3456(.5) 5 (.5)5 776.76.5 n
MA 5, Calculus II, Final Exam Preview, Spring 5 9 (d) 99 3456(+.5) (+.5) n 3456(.5) (.5) 73957.4.5 n. Determine whether or not the following series converges: n n n n 3.5 + n + Looking at the biggest degrees on top and the bottom, we decide to compare the given series to. There are three things to check: n.5 Are both n n n 3.5 + n and positive? Yes, all the leading coefficients, of n.5 n, n 3.5 and n.5, are positive. Is a n c n? Yes: n n? n 3.5 + n n.5 n.5 (n n )? n 3.5 + n n 3.5 n.5 n.5? n 3.5 + n n.5 n.5? n converges by the p-series test. n.5 Therefore, by Comparison Test, the n. Determine whether or not the following series converges: n n n 3.5 + n + converges. n sin(/n) As you try to size this one up, you should be a little confused, what the heck does n sin(/n) look like? Recall that for n large, we have /n close to, and for x close to, we have sin(x) x. Thus, for n large we have sin(/n) /n, and so n sin(/n). This should put you on the right track: 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. n
MA 5, Calculus II, Final Exam Preview, Spring 5 3. Determine whether or not the following series converges: As we size this one up, we see that e /n e, and so it looks like. There are three things to check: n n e /n n Are both e/n n and n positive? Yes, e/n, and n are both positive. e /n n? /n n e /n? n e /n? converges, by the p-series test, since p >. n Therefore, by the Comparison Test, e /n 4. Determine whether or not the following series converges: n converges. n! e n n As you try to size this one up you... well, maybe this isn t too easy to size up!! Just try the ratio test: (n+)! e lim (n+) n n! lim (n + )! en n e (n+) n! e n (n + )! e n lim n n! e (n+) e n lim (n + ) n e n +n+ lim n + n e n+ for L Hospital s L.H. lim n e n+ Therefore, by the Ratio Test, the series n! e n converges.
MA 5, Calculus II, Final Exam Preview, Spring 5 5. 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! 4 e x e c 4 e/4! e x e + e(x ) + e! (x ) + e 3! (x )3 + e 4! (x )4 +... n (x )n e n! 6. Find the center and radius of convergence. n n n (x + 3) n n n Radius of convergence: (x+3)n+ n+ n+ L lim n lim n (x+3)n n (x + 3) n + x + 3 Setting this to be < gives whence R. x + 3 < x + 3 < The center is 3, so we know that the series converges at least on the points ( 5, ). 7. (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.
MA 5, Calculus II, Final Exam Preview, Spring 5 (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) 3 3.9 y.9(x ) +.73 θπ/3 (b) XT cos(t )( + cos(t )) Y T sin(t )( + cos(t )) XT T Y T.9(T ) +.73
MA 5, Calculus II, Final Exam Preview, Spring 5 3 8. Find the area of the shaded region r + cos(θ)
MA 5, Calculus II, Final Exam Preview, Spring 5 4 A π π π r dθ ( + cos(θ) dθ + cos(θ) + cos (θ) dθ 3 4 + sin(θ) + 8 sin(θ) π 3 4 π 9. 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.
MA 5, Calculus II, Final Exam Preview, Spring 5 5 I II III IV V
MA 5, Calculus II, Final Exam Preview, Spring 5 6. 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 3.366 y 3.366 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:
MA 5, Calculus II, Final Exam Preview, Spring 5 7. Find the exact length of the curve AL 3 3 3 3 3 3 x e t + e t, y 5 t, t 3 (dx/dt) + (dy/dt) dt (et e t ) + 4 dt e t + e t + 4 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