MTH 133 Solutions to Exam 1 February 21, Without fully opening the exam, check that you have pages 1 through 11.

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1 MTH Solutions to Eam February, 8 Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the eam, check that you have pages through. Show all your work on the stanar response questions. Write your answers clearly! Inclue enough steps for the graer to be able to follow your work. Don t skip limits or equal signs, etc. Inclue wors to clarify your reasoning. Do first all of the problems you know how to o immeiately. Do not spen too much time on any particular problem. Return to ifficult problems later. If you have any questions please raise your han. You will be given eactly 9 minutes for this eam. Remove an utilize the formula sheet provie to you at the en of this eam. ACADEMIC HONESTY Do not open the eam booklet until you are instructe to o so. Do not seek or obtain any kin of help from anyone to answer questions on this eam. If you have questions, consult only the proctor(s). Books, notes, calculators, phones, or any other electronic evices are not allowe on the eam. Stuents shoul store them in their backpacks. No scratch paper is permitte. If you nee more room use the back of a page. You must inicate if you esire work on the back of a page to be grae. Anyone who violates these instructions will have committe an act of acaemic ishonesty. Penalties for acaemic ishonesty can be very severe. All cases of acaemic ishonesty will be reporte immeiately to the Dean of Unergrauate Stuies an ae to the stuent s acaemic recor. I have rea an unerstan the above instructions an statements regaring acaemic honesty:. SIGNATURE Page of

2 MTH Solutions to Eam February, 8 Stanar Response Questions. Show all work to receive creit. Please BOX your final answer.. (7 points) King Kong is fishing for airplanes atop the Empire State Builing using an elevator cable that weighs lb/ft. He catches one that weighs 4 lbs. How much work oes it take King Kong to reel in the airplane, raising it 8 ft in the process? Solution: We can set up a coorinate ais with the origin at the tip of Kong s fishing pole. We then note that. Each y ft section of the cable weights y lbs. Each section of cable at y ft below the tip of the pole nees to be lifte y ft.. The plane itself will nee 4 8 =,, ft-lbs to be lifte to the tip of the pole. So, the total work is,, + 8 y y ft-lbs =, 84, ft-lbs. Let R be the region boune by the curves y = ln(/), =, y =, an y =. (a) ( points) Sketch the region R; make sure to label the -intercept(s) an shae the region R. Solution: Check for:. the -intercept at =,. for the basic shape of ln() as a concave function, an. for the properly shae region. (b) (4 points) Set-up, but o not evaluate, an integral representing the volume of the soli forme by revolving R aroun the y-ais. Solution: At position y along the y-ais we integrate from = to = e y. Thus, the integral is π(e y ) y Page of

3 MTH Solutions to Eam February, 8. (7 points) Let f() = + + π sin(π). Knowing that f() =, what is the value of (f ) ()? Solution: Using the formula sheet we have that (f ) () = f (f ()) = f (). Since f () = + + cos(π) we can also see that f () = 5 = 4. As a result we have that (f ) () = (7 points) Compute the erivative of the function g() = ( + cosh() ) +. Solution: We have that g() = e ln(+cosh())(+) so that g () = e ln(+cosh())(+) (ln( + cosh())( + )) ( ) + = e ln(+cosh())(+) + cosh() (cosh()) + ln( + cosh()) ( ) + = e ln(+cosh())(+) sinh() + ln( + cosh()) + cosh() = ( + cosh() ) ( ) + + sinh() + ln( + cosh()) + cosh() Page of

4 MTH Solutions to Eam February, 8 5. Evaluate the following integrals. Show all work. (a) (7 points) sin cos 4 Solution: sin cos 4 = = sin ( cos ()) cos 4 sin (cos 4 cos 6 ) Letting u = cos(), u = sin an integrating we get sin cos 4 = u 6 u 4 u (b) (7 points) +. = 7 u7 5 u5 + C = 7 cos7 5 cos5 + C Solution: We first factor the enominator to get that + = ( + )( ). We now use partial fractions to set that + = A + + B where A( ) + B( + ) =. This means that A + B = an B A = both hol, which implies that A = B = an B =. As a result, + = ( + ) + ( ) + C = (ln( + ) ln( )) + C = ( ) ln + + C Page 4 of

5 MTH Solutions to Eam February, 8 6. (7 points) Evaluate the integral 5. Solution: We nee to notice that this is an improper integral. 5 = lim a a a = lim a lim b + b + lim b + 5 b + = lim [ + ln ]a a + lim [ + ln ]5 b + b = lim a + ln a ln + lim 5 + ln b ln b a b + This DNE, however, since lim a ln a = an lim b + ln b =. 7. (7 points) Solve the initial value problem y = 9y with initial value y( ) =. Solution: The equation is separable. We have that y = 9y y = (y) Taking the initial value into account we see that sin (y) = + C y = sin( + C) y = sin( + C) = sin( + C) = + C = nπ = C = + nπ where n is any integer. Thus, y() = sin( + + nπ) for any integer n. Note: We only epect them to report one value of n that works, e.g., n =. Page 5 of

6 MTH Solutions to Eam February, 8 Multiple Choice. Circle the best answer. No work neee. No partial creit available. 8. (4 points) Evaluate the limit lim + A.. B.. C.. ( ) ln. D. The limit oes not eist: it tens to. E. The limit oes not eist: it tens to. ( 9. (4 points) Evaluate the limit lim tan () π ). A.. B.. C.. D. The limit oes not eist: it tens to. E. The limit oes not eist: it tens to.. (4 points) Which statement is FALSE? A. B. C. D. E. 5 converges. iverges. iverges. ln converges. iverges. Page 6 of

7 MTH Solutions to Eam February, 8. (4 points) The population in a bacterial culture grows eponentially. A culture starte with cells. After 5 hours cells were observe. After how many aitional hours will there be 6 cells? 5 A.. B ln C. ln. D.. E. 5 ln ln. ( (. (4 points) cos tan 5 ) ) =? A. 5 B. C. D. E (4 points) Choose the correct partial fraction ecomposition of A B. + + C. + + D. + E Page 7 of

8 MTH Solutions to Eam February, 8 4. (4 points) Evaluate the efinite integral 4 A. B. C. D. E. π 6 π 8 π 4 π π π/ sin (). 5. (4 points) Evaluate the efinite integral A. B. π C. 6 D. π 8 π 4 E. π π / cos(π). 6. (4 points) Let f() = cos() + ( +. Which of the following statements is correct concerning the improper ) integral f()? A. Since f(), by the comparison test the integral iverges. 4 B. Since f(), by the comparison test the integral converges. C. Since f(), by the comparison test the integral converges. D. Since f(), by the comparison test the integral iverges. E. The comparison test cannot be use because f() changes signs. Page 8 of

9 MTH Solutions to Eam February, 8 More Challenging Questions. Show all work to receive creit. Please BOX your final answer. 7. (7 points) Suppose that a function f has f() for all >, eplain why the integral f() 8. Solution: We want to use the Comparison Theorem for Integrals from Section 7.8. Appealing to that along with our knowlege that f() for all > we have f() f() lim a a f() lim a [ ] a f() lim a a + 8 f() (7 points) Using what you learne about inverse functions, fin a formula for the erivative sinh (). Your answer shoul be epresse in a way that oes not use hyperbolic functions or their inverses. Solution: We can begin by taking the erivative of both sies of our basic inverse function relationship, e.g., that sinh ( sinh () ) =. Taking the erivative of both sies, an remembering to use the chain rule, we get that cosh ( sinh () ) sinh () = = sinh () = cosh ( ). sinh () () To get ri of the hyperbolic functions we now nee to use that cosh () sinh () = from the formula sheet. Rearranging this we learn that cosh() = + sinh (). As a result we have that cosh ( sinh () ) = Plugging this into () we learn that + sinh (sinh ()) = + sinh () = + Page 9 of

10 MTH Solutions to Eam February, 8 DO NOT WRITE BELOW THIS LINE. Page Points Score Total: 6 No more than points may be earne on the eam. Page of

11 MTH Solutions to Eam February, 8 Integrals FORMULA SHEET Derivatives Volume: Suppose A() is the cross-sectional area of the soli S perpenicular to the -ais, then the volume of S is given by (sinh ) = cosh Inverse Trigonometric Functions: (cosh ) = sinh V = b a A() (sin ) = (csc ) = Work: Suppose f() is a force function. The work in moving an object from a to b is given by: W = b = ln + C tan = ln sec + C a f() sec = ln sec + tan + C a = a ln a + C for a Integration by Parts: u v = uv v u (cos ) = (tan ) = (sec ) = + (cot ) = + If f is a one-to-one ifferentiable function with inverse function f an f (f (a)), then the inverse function is ifferentiable at a an (f ) (a) = f (f (a)) Hyperbolic an Trig Ientities Hyperbolic Functions sinh() = e e cosh() = e + e csch() = sinh sech() = cosh tanh() = sinh cosh coth() = cosh sinh cosh sinh = cos + sin = sin = ( cos ) cos = ( + cos ) sin() = sin cos sin A cos B = [sin(a B) + sin(a + B)] sin A sin B = [cos(a B) cos(a + B)] cos A cos B = [cos(a B) + cos(a + B)] Page of