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 Page of
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 4000 lbs. How much work oes it take King Kong to reel in the airplane, raising it 800 ft in the process?. Let R be the region boune by the curves y = ln(/), = 0, y =, an y =. (a) (3 points) Sketch the region R; make sure to label the -intercept(s) an shae the region R. y 3 4 5 6 7 (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. Page of
3. (7 points) Let f() = 3 + + π sin(π). Knowing that f() = 3, what is the value of (f ) (3)? 4. (7 points) Compute the erivative of the function g() = ( + cosh() ) 3+3. Page 3 of
5. Evaluate the following integrals. Show all work. (a) (7 points) sin 3 cos 4 (b) (7 points) +. Page 4 of
6. (7 points) Evaluate the integral 5 0. 7. (7 points) Solve the initial value problem y = 9y with initial value y( ) = 0. Page 5 of
Multiple Choice. Circle the best answer. No work neee. No partial creit available. 8. (4 points) Evaluate the limit lim 0 + A. 0. 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. 0. B.. C.. D. The limit oes not eist: it tens to. E. The limit oes not eist: it tens to. 0. (4 points) Which statement is FALSE? A. B. C. D. E. 5 0 3 0 converges. iverges. iverges. ln converges. iverges. Page 6 of
. (4 points) The population in a bacterial culture grows eponentially. A culture starte with 000 cells. After 5 hours 3000 cells were observe. After how many aitional hours will there be 6000 cells? 5 A.. B. 5. 5 ln 3 C. ln. 0 D. 3. 5 ln E. ln 3. ( (. (4 points) cos tan 5 ) ) =? A. 5 B. C. D. E. 3 5 3 3 3 5 3. (4 points) Choose the correct partial fraction ecomposition of A. + + + B. + + C. + + D. + E. + + 3 4. Page 7 of
4. (4 points) Evaluate the efinite integral 4 A. 0 B. C. π 6 D. π 8 3π 4 E. π π π/ 0 sin (). 5. (4 points) Evaluate the efinite integral A. 0 B. C. π 6 D. π 8 3π 4 E. π π / 0 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
More Challenging Questions. Show all work to receive creit. Please BOX your final answer. 7. (7 points) Suppose that a function f has 0 f() for all > 0, eplain why the integral 3 0 3 f() 8. 8. (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. Page 9 of
DO NOT WRITE BELOW THIS LINE. Page Points Score 4 3 4 4 4 5 4 6 7 8 9 4 Total: 06 No more than 00 points may be earne on the eam. Page 0 of
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