MTH 133 Solutions to Exam 1 October 10, Without fully opening the exam, check that you have pages 1 through 11.

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1 MTH 33 Solutions to Eam October 0, 08 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 90 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 α Page of

2 MTH 33 Solutions to Eam October 0, 08 Stanar Response Questions. Show all work to receive creit. Please BOX your final answer.. Evaluate the following integrals. Show all work. (a) (7 points) + 9 Use trig substitution with 3 = tan(θ), 3 = sec (θ) θ, + 9 = + (3) = sec (θ). Integral becomes = sec (θ) 3 sec (θ) θ = 3 θ = 3 θ + C = 3 tan (3) Alternatively: use u substitution with u = 3 an use the formula sheet which shows tan (u) + C. + u u = (b) (7 points) cosh(). Use integration by parts, with u = an v = cosh() to get u = v = sinh() u = v = cosh() an cosh() = uv v u = sinh() sinh() = sinh() cosh() + C. Page of

3 MTH 33 Solutions to Eam October 0, 08. Evaluate the following limits. (If you use L Hopital s Rule, eplicitly state your reasoning.) ( ) (a) (7 points) lim / ln ( ) lim ln / = lim ln () First simplify (= ) Check is ineterminate form L H = lim = lim = 0 = lim L Hopital s rule Simplify (b) (7 points) lim y 0 cos(4y) e 8y cos(4y) lim y 0 e 8y (= = 0 ) Check is ineterminate form 0 L H = lim y 0 4 sin(4y) 8e 8y = L Hopital s rule = 0 Compute the limit Page 3 of

4 MTH 33 Solutions to Eam October 0, (7 points) A 500 lb anchor hangs off of an airship from a chain that is 00 ft long. The chain itself weighs another 00 lbs (in aition to the weight of the anchor). How much work will it take to pull the anchor up onto the eck of the airship by the chain?. Density of the chain to is 00lbs 00ft = lbs/ft. Work one to pull up anchor itself: 500 lbs 00 ft = 50,000 ft-lbs. 3. Work one to pull up chain: Total work = 60,000 ft-lbs. 0 (00 y) y = 00y y 00 = = Let R be the region boune by the curves y =, y =, y =, an the y-ais. (a) ( points) Sketch the region R; make sure to shae an label the region R. y R (b) (5 points) Fin the volume of the soli forme by revolving R aroun the y-ais.. The area of a thin slice of the soli of rotation at height y is π = π(y ).. The volume of the thin slice is πy 4 y. 3. The total volume is therefore = 4. Final answer is 3π/5 πy 4 y. Page 4 of

5 MTH 33 Solutions to Eam October 0, (7 points) Give the partial fraction ecomposition of Factor the enominator = ( + )( + ). Setup for partial fraction ecompostion 3. Cross multiply gets this nees to be equal to the numerator. ( + )( + ) = A + + B + A( + ) + B( + ) = (A + B) + (A + B) 4. So solve an get 5. So that A + B = A + B = 0 A = an B = ( + )( + ) = (7 points) Evaluate the integral. Treat improper integral with limits. Compute the integral lim a a Simplify the limit using rules of logarithm a = lim a + = lim [ln(a ) ln(a + ) ln() + ln(3)] a lim [ln(a ) ln(a + ) 0 + ln(3)] = ln(3) + ln a ( lim a ) a a + 4. Compute the limit = ln(3) + ln() = ln(3) Page 5 of

6 MTH 33 Solutions to Eam October 0, 08 Multiple Choice. Circle the best answer. No work neee. No partial creit available. 7. (4 points) Evaluate the integral A. B. + C C. + C D. + C + C E. + C ( ( 8. (4 points) sin tan 5 ) ) =? A. 5. B. C. D. E (4 points) The erivative of f() = cos() is A. sin() ln + cos() B. sin() ln + cos() C. D. E. ( sin() ln + cos() ) cos() ( sin() ln + cos() ) ( sin() ln + cos() ) cos() + C cos() Page 6 of

7 MTH 33 Solutions to Eam October 0, (4 points) Solve the initial value problem y A. y = ln( ). = y with initial value y(0) = 0. B. y = ln( ). C. y = e. D. y = e. E. y = e.. (4 points) Evaluate A. /40 B. /4 C. 0 D. /4 E. /40 π 0 sin 7 cos 3.. (4 points) Compute f () if f() = tan (). A. tan () ln() sec () B. tan () ln + C. tan () ln + D. tan () + E. tan () + Page 7 of

8 MTH 33 Solutions to Eam October 0, (4 points) 5 ays ago, a sample of raioactive substance is measure at 00 grams. Toay, the sample measures at 00 grams. How many ays from now will there be only 5 grams of the sample remaining? A B. 5 C. 7.5 D. 0 E (4 points) Let f() = e +. Which of the following statements is correct concerning the improper integral f()? A. Since f(), by the comparison test the integral converges. B. Since f(), by the comparison test the integral iverges. C. Since f() 3, by the comparison test the integral converges. D. Since f() + e, by the comparison test the integral converges. E. The comparison test cannot be use. 5. (4 points) If f() = 4 + cos(), fin (f ) (), knowing that f(0) =. A.. B. 4. C. 3. D. 4. E.. Page 8 of

9 MTH 33 Solutions to Eam October 0, 08 More Challenging Questions. Show all work to receive creit. Please BOX your final answer. 6. (7 points) You an your frien both evaluate an integral an get ifferent answers. You get that the integral is tan ()/+C, an your frien gets that the integral is sec ()/+C. Your TA tells the two of you that you are both correct. Use the constant of integration to eplain how this can be the case even though tan() sec(). The two functions sec () an tan () iffers by a constant (in fact ) ue to the trig ientity sec () = tan () + Since the constant of integration is arbitrary, both answers are correct. 7. (7 points) Eplain why start by thinking about how sin( ) relates to sin(). sinh(sin()) = 0. Hint: Don t try to evaluate this integral. Instea, the function sin() is an o function. the function sinh() is an o function. the composition of o functions is still o: so sinh(sin()) is an o function. integrating any o function on an interval ( a, a) is always zero. Page 9 of

10 MTH 33 Solutions to Eam October 0, 08 DO NOT WRITE BELOW THIS LINE. Page Points Score Total: 06 No more than 00 points may be earne on the eam. Page 0 of

11 MTH 33 Solutions to Eam October 0, 08 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)) 0, 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