Math 19 Final Exam – March 14thName: SUID#:• Complete the following problems. In order to receive full credit, please show all of your work andjustify your answers. You need to include the units in your answers when relevant.• You do not need to simplify your answers unless specifically instructed to do so. You may use anyresult from class that you like, but if you cite a theorem be sure to verify the hyp otheses are satisfied.• You have 3 hours. This is a closed-book, closed-notes exam. No calculators or other electronicaids will be permitted. If you finish early, you must hand your exam paper to a member of teachingstaff.• Please check that your copy of this exam contains 15 numbered pages and is correctly stapled.• If you need extra room, use the back sides of each page. If you must use extra paper, make sure towrite your name on it and attach it to this exam. Do not unstaple or detach pages from this exam.• Please sign the following:“On my honor, I have neither given nor received any aid on thisexamination. I have furthermore abided by all other aspects of thehonor code with respect to this examination.”Signature:The following boxes are strictly for grading purposes. Please do not mark.Question: 1 2 3 4 5 6 7 8 9 10 11 12 TotalPoints: 9 10 9 8 14 23 14 13 7 9 15 11 142Score:Math 19, Winter 2011 Final Exam – March 14th Page 1 of 151. (9 points) Find each of the following limits, with justification (show steps). If there is an infinite limit,then explain whether it is ∞ or −∞.(a) (5 points) limx→3(x − 1)(x − 2) − 2x2− 5x + 6(b) (4 points) limx→∞x4+ 2x2− 9e−xMath 19, Winter 2011 Final Exam – March 14th Page 2 of 152. (10 points) Careful: These two questions are unrelated.(a) (3 points) State the extreme value theorem.(b) (7 points) Find the derivative of f (x) =√1 − 2x using the definition. No marks will be awardedif you get the right answer using other methods.Math 19, Winter 2011 Final Exam – March 14th Page 3 of 153. (9 points) Careful: These two questions are not related.(a) (5 points) Find the inverse function of f(x) =11 − 2 ln(x)(b) (4 points) Find the value/values of a (a number) that make the function g(x) =ax2+ ax − 3 if x ≤ 2ax2+2if x > 2continuous at x = 2.Math 19, Winter 2011 Final Exam – March 14th Page 4 of 154. (8 points) Mark each statement below as true or false by circling T or F. No justification is necessary.T F If f(1) = 3 and f (x) is one-to-one, then x = 3 is in the domain of f−1(x)T F If a function has a vertical tangent line at x = 1, then it is not continuous at x = 1T F Rational functions are continuous everywhereT F The function f(x) = e(x3)is increasing at x = −8T F If limx→∞f(x) = 0, and g(x) is another function, then limx→∞f(x)g(x) = 0T F If f00(40) = 0, then f(x) has an inflection point at x = 40T F A function can not have an infinite number of local maximaT F Assuming f(x) is a differentiable function, the derivative of the function f (√x) isf0(√x)2√xMath 19, Winter 2011 Final Exam – March 14th Page 5 of 155. (14 points) Water is flowing into a bathtub. B ut the bathtub has a hole somewhere, so water is leakingout too. At time t (in seconds), the volume of water in the tub is V (t) = e−t2+6tliters. Please includethe units in the answers to the following questions.(a) (4 points) Compute the (instantaneous) rate of change of the volume at time t = 2 and at timet = 6.(b) (4 points) Interpret the sign of your answers to part (a) in terms of water flowing in and waterleaking out at those times.(c) (6 points) Find limt→∞V (t) and interpret this result in terms of time and volume of water in thebathtub. (Hint to do the limit: Do limt→∞(−t2+ 6t) first.)Math 19, Winter 2011 Final Exam – March 14th Page 6 of 156. (23 points) Consider the function f (x) =x2x + 3(a) (2 points) Find the domain of f(x).(b) (5 points) Where is f(x) increasing? And decreasing? Find the x-coordinates of all the localmaxima and minima of f(x).Math 19, Winter 2011 Final Exam – March 14th Page 7 of 15For convenience, here is the function again: f (x) =x2x + 3(c) (3 points) It is known that f00(x) =18(x + 3)3. Where is f (x) concave up? And concave down?Find the x-coordinates of all the inflection points of f (x).(d) (5 points) Find all the asymptotes of f(x). If there are any vertical asymptotes, find the sidelimits.Math 19, Winter 2011 Final Exam – March 14th Page 8 of 15For convenience, here is the function again: f (x) =x2x + 3(e) (2 points) Find the y-intercept of f(x) and the x-intercepts (if there are any).(f) (6 points) Sketch the graph of f(x) in the axes provided below. Your graph should be faithful tothe information you obtained in the previous parts of this problem.Math 19, Winter 2011 Final Exam – March 14th Page 9 of 157. (14 points) Find the derivatives of the following functions using any method you like (that works).You do not need to simplify your answers.(a) (4 points) f (x) = sin(e + ln(sin(x)))(b) (4 points) g(t) = [t + arctan(t)]2p3t2+ 1(c) (6 points) Z(x) =1x(1x)Math 19, Winter 2011 Final Exam – March 14th Page 10 of 158. (13 points) Linear approximation.(a) (8 points) Estimate the value ofp(1.1)3+ 15(1.1) using linear approximation. Simplify youranswer. A quotient of two integers or a decimal number are both valid answers.(b) (5 points) Is your e stimate smaller or greater than the real value? Justify your answer.Math 19, Winter 2011 Final Exam – March 14th Page 11 of 159. (7 points) A company produces toys. The cost of producing n toys a month is C(n) =300ndollars.Each toy is sold for 25 dollars.(a) (1 point) Assuming every toy produced is sold, express the profit as a function of the monthlyproduction n.(b) (6 points) Some guy is hired to count the toys in the production line and compute the profitbeforehand using this information. This month he counted 100 toys, but he fell asleep some-times, so his count may be off by 10 toys. Using differentials, estimate the possible error in thiscomputation of the profit. Give your answer in dollars and cents.Math 19, Winter 2011 Final Exam – March 14th Page 12 of 1510. (9 points) Implicit differentiation. Consider the curve√x + 3y sin(y) = 2. Find the equations of thetangent lines to the curve at the points (4, 0) and (4, π).Math 19, Winter 2011 Final Exam – March 14th Page 13 of 1511. (15 points) We want to make a book. Each page must contain 32 in2of print, and must have 2 inchesmargins at the
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