Math 2 (Calculus I) Final Eam Form A KEY Multiple Choice. Fill in the answer to each problem on your computer-score answer sheet. Make sure your name, section an instructor are on that sheet.. Approimate 5 4 using a Left Han sum with 2 subintervals (n=2). (a) 82 (b) 64 (c) 8 () 62 (e) 624 (f) 625 (g) None of these 2. Fin the area uner the function f() = from = to = 8. (a) 45 4 (b) 4 (c) 2 () 5 (e) 2 (f) None of these. Given the limit statement lim (2 ) = pick the largest δ that works with the efinition of the limit if ɛ = 0.06. (a) 0.00 (b) 0.005 (c) 0.0 () 0.02 (e) 0.0 (f) No such δ eists 4. Which of the following is an inflection point of f() = 2 +? (a) (b) (c) 2 () 2 (e) 2 (f) 2 (g) (h) 5. Given ln y y ln = e 2 2e, fin y at the point (e2, e). (a) 0 (b) e (c) e 2 () e e 2 (e) e e 2 2e (f) e 2 2e (g) e e 2 6. Which of the following are -values for which f() = sin() has a local maimum? (a) 2π (b) π (c) 0 () π (e) 2π (f) More than one of these (g) None of these 7. Which of the following functions has a iscontinuous first erivative? (a) sinh() (b) / (c) tan () () (e) ln( 2 + ) + 2 (f) All of the first erivatives of these functions are continuous 8. 2 + t t =
a) + (2) 2 b) 2 + (2) 2 c) + 2] ) 2 + 2 e) + (2) f) 2 + (2) g) + h) 2 + f) Short Answer: Fill in the blank with the appropriate answer. 9. ( points) (a) Simplify sin ( ( )) cos = 4 5 5 (b) 2 e = e + C (c) cos() (ln(sin )) = sin() = cot() () (e) (sinh2 ()) = 2 sinh() cosh() or sinh(2) (e + ) = e + 2 (f) If f () = e + sin + 2, then f() = e cos() + + C (g) lim tan () = π 2 (h) lim 2 + 2 2 4 = Does not eist (i) lim sin ln = 0 + (j) 2 4 + 5 = 2 2 (k) (2 ) = ln(2)2
Free response: Write your answer in the space provie. 0. (8 points) (a) If f() =, use the efinition of a erivative to set up a limit to fin f (). f () h 0 +h h (b) Fin f () by evaluating the limit. (No points will be aware if ifferentiation rules are use.) f () h 0 +h h h 0 h 0 ( + h) =. 2 h h( + h). (6 points) Fin the imension of the largest rectangle that can be inscribe between the curve y = 4 2 an the -ais. The rectangle will have corners at (, 0), (, 0), (, 4 2 ), an (, 4 2 ). Thus, the with of the rectangle is 2 an the height is 4 2. The area therefore is A() = 2(4 2 ) = 8 2. Notice that A () = 8 6 2. If we set the erivative to 0, we have 2 = 4, or = 2. Thus, the imensions are ( 4, 8 ). Here the area is A() = 2(9 2 ) = 8 2. A () = 8 6 2 = 0 gives 2 =, so =. Thus, the imensions are (, 6). 2. (4 points) lim 0 ln() sin() Use L Hopital s rule: lim 0 ln lim ln() sin() 0 0 csc. ln csc 0 csc() cot() sin tan 0 0 cos tan + sin sec 2 = 0. cos() lim ln()( cos()) 0 0 ln sin() cos() 0 = ln 2 + 2 ln = 0. ln 2
. ( points) 4. (4 points) ( ( ln e sin )) 2 + 4 ( ) e sin Let u = 2 + 4. Then u = 2 an 2 + 4 = 2 Let u = + 9. Then u = 2 an 2 ( e + e ) cos sin 2 u u = ln u + C 2 2 ln(2 + 4) + C. + 9 = u u = ln u + C = ln + 9 + C.
5. (0 points) Give the following information about the function f() = 4 4 : (If no information is available in a particular category, leave it blank or cross it out. Putting information in where none eists will be treate as an incorrect answer). All -intercepts = (0, 0) (4, 0) y-intercept =(0, 0) Intervals for which f() is increasing: (, ) Intervals for which f() is ecreasing: (, ) Coorinates of all inflection points: (0, 0) (2, 6) Intervals for which f() is concave up: (, 0), (2, ) Intervals for which f() is concave own: (0, 2) Coorinates of any local maimums: None Coorinates of any local minimums: (, 27) Graph the function: Graph the function: All -intercepts = (0, 0) (5, 0) y-intercept =(0, 0) Intervals for which f() is increasing: (, 0) (4, ) Intervals for which f() is ecreasing: (0, 4) Coorinates of all inflection points: (, 62) Intervals for which f() is concave up: (, ) Intervals for which f() is concave own: (, 0), (0, ) Coorinates of any local maimums: None Coorinates of any local minimums: (4, 256) 6. (6 points) A certain element has a half life of 20 years. How many years will it take until only ( ( ) ( ) 0% of the element remains? Note: ln.7 an ln 2.. You can either leave 2 0 ) your answer in terms of logs or give a numerical answer using these approimations. Since 2 = e20r, it is easy to see that r = ln(/2) 20 we nee. = e ln 2 20 t, or Hence, ln(.) = ln 2 20 t. t = ln 2 20 ln(.) = ln 2 20 ln(0). = ln 2. For 0 percent to remain, 20
7. (6 points) The equation of the tangent line to the curve y = ( at 2, ). 2 4 Note that y = 2. So, y (2) = 4. Thus, y 4 = ( 2), 4 an y = 4 + 4. y = 4. y ( 2 ) = 48. y 8 = 48( 2 ) or y = 48 + 2. 8. (6 points) Use linear approimation to estimate 6: Let f() =. Then f () = 2. 6 = 64 + 2 64 ( ) 8 6 = 27 6. 9. (6 points) A pump is blowing up a spherical balloon with a pump rate of 0cm ( /sec. How fast is the iameter of the balloon growing when the balloon has a 5cm raius? Volume of a sphere is given by 4 ) πr. Differentiate V = 4 πr to get Thus, an V t = 4πr2 r t. 0 = 4π 25 r t, r t = 0π. 0 = 4π 9 r t r t = 5 8π.
20. (6 points) A particle is moving with the given ata. Fin the position function of the particle. a(t) = sin(t) + cos(t), s(0) = 0, v(0) = 2. so C =. Thus, v(t) = cos(t) + sin(t) + C v(0) = + C = 2 v(t) = cos(t) + sin(t) +. s(t) = sin(t) cos(t) + t + D s(0) = + D = 0, D = s(t) = sin(t) cos(t) + t +. END OF EXAM