Math 121 - Final Exam - 12/12/2015 Name: Section: Section Class Times Day Instructor Section Class Times Day Instructor 1 09:00AM -09:50AM M T W F Sarah G Rody 12 11:00 AM - 11:50 AM M T W F Marci Ann Perlstadt 2 09:00AM -09:50AM M T W F Huilan Li 13 04:00 PM - 04:50 PM M T W F Ronald K Perline 3 09:00AM -09:50AM M T W F Hwan Yong Lee 18 12:00 PM - 01:50 PM M W Amanda Grace Lohss 4 02:00PM-02:50PM M T W F Ronald K Perline 19 02:00 PM- 02:50 PM M T W F Dimitrios Papadopoulos 6 10:00 AM - 10:50 AM M T W F Marci Ann Perlstadt 20 01:00 PM - 01:50 PM M T W F Dimitrios Papadopoulos 7 10:00 AM - 10:50 AM M T W F Huilan Li 21 01:00 PM - 01:50 PM M T W F Huilan Li 8 12:00 PM- 12:50 PM M T W F Hwan Yong Lee 22 05:00 PM - 05:50 PM M T W F Dimitrios Papadopoulos 10 01:00 PM - 01:50 PM M T W F Kenneth P Swartz 23 11:00 PM - 11:50 PM M TW F Hwan Yong Lee The following rules apply: This is a closed-book exam. You may not use any books or notes on this exam. Page Points Score For free response questions, you must show all work. Answers without proper justification will not receive full credit. Partial credit will be awarded for significant progress towards the correct answer. Cross off any work that you do not want graded. For multiple choice questions, circle the let- ter of the best answer. Make sure your circles include just one letter. These problems will be marked as correct or incorrect; partial credit will not be awarded for problems in this section. 2 3 4 5 6 8 10 10 10 10 15 6 You have 2 hours to complete this exam. When time is called, stop writing immediately and turn in your exam to the nearest proctor. 9 10 6 6 You may not use any electronic devices including (but not limited to) calculators, cell phone, or ipods. Using such a device will be considered a violation of the university's academic integrity policy and, at the very least, will result in a grade of 0 for this exam. 11 12 13 6 6 6 For word problems, all answers must include the appropriate units in order to earn full credit. 14 15 6 3 Total: 100
Math 121 Final Exam- Page 2 of 15 12/12/2015 Part I: Free Response 1. (10 points) Let f(x) = vr=-x. Use the definition of the derivative to find f'(x). --J\ - X- - \,1\A - "'-~o :::: \\IV\ -\ \.-.,. ~D
Math 121 Final Exam- Page 3 of 15 12/12/2015 2. (10 points) Let Use logarithmic differentiation to find f'(x). -\ \ -::; y
Math 121 Final Exam - Page 4 of 15 12/12/2015 3. (10 points) Evaluate the following limit. If the limit does not exist, write +oo, -oo, or DNE (whichever is most appropriate). lim (esc x -.!_) ------? co - ()(') x--+o+ x \ +
Math 121 Final Exam- Page 5 of 15 12/12/2015 4. (10 points) A farmer has 240 meters of wire fence with which he plans to build two adjacent rectangular pens. What are the dimensions of the enclosure that has the maximum area? For full credit, you must identify the interval over which you are maximizing and include units in your final answer..., Y------1 A A f:: \ LO-,., b 0
Math 121 Final Exam- Page 6 of 15 12/12/2015 5. (15 points) Consider the given information: x2 f(x)= ~ X - 4! "( ) = 8(3x2 + 4) x (x2-4)3 (a) Determine the x-intercepts and the y-intercepts. Write "None" if there are no such intercepts. 0 (b) Determine the equations of any horizontal and vertical asymptotes. Write "None" if there are no such asymptotes. -0 (c) Determine the interval(s) on which f(x) is increasing and those on which f(x) is decreasing. L ~ IS O.l\ \) I 1\.
Math 121 Final Exam- Page 7 of 15 12/12/2015 (d) Determine the interval(s) on which f(x) is concave up and those on which f(x) is concave down. 0 CH\ (-CO I - 2_) caaj a.-a (:2-,, LJ. (e) Identify the coordinates ( x, y) of all local extrema (maxima or minima) and all inflection points. If there are none, write "None." ";:. 0. (f) On the axes provided below, sketch the graph of the f ( x). Label the.coordinates of all critical points, inflection points, and intercepts with the coordinate axes. Also, label all horizontal asymptotes and vertical asymptotes, if any. 3 4 5
~ L Math 121 Final Exam- Page 8 of 15 12/12/2015 Part II: Multiple Choice 6. (3 points) Let f(x) = s~n kx' if x =1-0 s1n2x \ k + 1, if X= 0 Find a value of k for which f(x) is continuous at x = 0. (a) 0 (b) 1 (c) -1 (d) 2 7. (3 points) Find the equation of the tangent line to f(x) = x 2 V5- x2 at x = 1. 17 9 (a) y=-x-- 4 4 17 1 (b) y =-X-- 4 4 - -x-- @= 7 3 2 2 7 5 (d) y = -x-- 2 2 1 3 (e) y=-x+- 2 2 y- "' f l :::: -z..... ( "'- y:._-.2. 1 -L-x
Math 121 Final Exam- Page 9 of 15 12/12/2015 )4x 2 + 2x- 1 8. (3 points) Evaluate the following limit: lim "- \ \M x-+-oo -2x + 2 (a) 1 @1 (c) 1-2 (d) 1 -- 2 (e) -00 9. (3 points) Let f(x) =3ft. Evaluate j'(1). (a) 1 (\ 2 (b) 3 (c) 3ln 3 (d) - ln3 2 3ln3 -- 2
Math 121 Final Exam- Page 10 of 15 12/12/2015 10. (3 points) Evaluate lim - 1 x x-to+ nx (b) 2 (c) 4 (d) -()() (e) +oo tan 15 (x +h)- tan 15 (x) 11. (3 points) Evaluate lim h h-+0 (a) 0 (d) 00 (e) none of the above
Math 121 Final Exam- Page 11 of 15 re 12/12/2015 (a) x=-1 (b) X= 0 (c) x = 1 (d) X= 2 (e) x = 3 / L ~/ ~"" / "" / / 13. (3 points) Let Evaluate dy when x = 0 and y = 1. dx (a) 0 (b) 1 + () + - 4 -:::.. 0 (c) 1 fl = ~ -- 4 (d) 1-3
Niath 121 Final Exam - Page 12 of 15 12/12/2015 14. (3 points) Find the local linear approximation L(x) for f(x) = cos- 1 (2x) at x = 0. (a) L( x) = 1 + 2x (b) L( x) = 1-2x 7r (c) L(x) = 2 + 2x ~L(x) = ~ - 2x. (e) L(x) = 2x y- h)<)~ i c~) 1 "? -.,.,., ~. :.. -7, Ce::>~..:. 0 -Lf ')(. l. ~ - - 15. (3 points) Assume x = x(t) andy = y(t). Suppose that x 3 - y 3 = 2 and ddx.= 2 at (1, -1). t Find ddy at (1, -1).. t (a) 0 (c) -2 2..- (d) 3 (e) -3
Math 121 Final Exam - Page 13 of 15 12/12/2015 16. (3 points) Evaluate lim ~.;,. x-ro In 1- x (a) 0 1 (b) -- 2 (c) ~ 2 2 - L (e) 2 17. (3 points) Evaluate lim(l- x)- 4 /x x-ro :_ \\IV"\ - v.. t..-1 \ l --y. \\~ (c) 4 ~ (d) -4 (e) 1
Math 121 Final Exam- Page 14 of 15 12/12/2015 18. (3 points) At which of the following x-value does f(x) = x 3 - minimum? (a) -5 i'c ~ )(; + D (W5..: (c) -2 'i X - '<- 'X_ :: '2._) s -- 21 x 2 + 30x- 3 have a relative 2 Q (d) 2 + (e) 3 \ 19. (3 points) Find the value of the absolute maximum of f(x) = si~x- cosx on [O,n]. (a) -1 ' 1 (x) C<Y>';:, (b) 0 c_., '\:,. y; -1::: 'S,'I\ )<:, --:-0 W' x::: ~ (d) V3 2 ~ ~ -1 (e) 2 ~( -: \ "')._ ~
Math 121 Final Exam- Page 15 of 15 12/12/2015 20. (3 points) Consider f(x) = xe-x. Which of the following are true? I. x = 0 and x = 1 are critical points of f(x) c:!!!:) f(x) is concave down on ( -oo, 2) and f(x) is concave up on (2, oo) II. f(x) is decreasing on (-oo, 1) and f(x) is increasing on (l,oo) ) f(x) has an inflection point (2, : 2 ) (a) I and II only (b) II and III only <@II and IV only (d) II, III, and IV only (e) I, II, III, and IV