EXAM 2 MATH 161 BLAKE FARMAN Lafayette College Answer the questions in the spaces provided on the question sheets and turn them in at the end of the exam period. It is advised, although not required, that you check your answers. All supporting work is required. Unsupported or otherwise mysterious answers will not receive credit. You may not use a calculator or any other electronic device, including cell phones, smart watches, etc. By writing your name on the line below, you indicate that you have read and understand these directions. Name: Solutions Problem Points Earned Points Possible 1 5 2 4 3 2 4 1 5 3 6 15 7 20 8 15 9 15 10 20 Total 100 Date: October 24, 2018. 1
2 EXAM 2 Fill In the Blank 1 (5 Points). Assume that f is a function such that f 0 (x) andf 00 (x) aredefinedforallx. (a) A point c is a critical point of f if flat o or is undefined (b) f is increasing on an interval if (c) f is decreasing on an interval if (d) f is concave up on an interval if (e) f is concave down on an interval if. 2 (4 Points). The first derivative test says that a critical point, c, off is: a local maximum if f 0 changes from positive to at c; negative local minimum if f 0 changes from to at c. 3 (2 Points). The second derivative test says that a critical point, c, off is: a local maximum if f 00 is at c; negative a local minimum if f 00 is at c. positive 4 (1 Point). Suppose that f 00 (c) =0. Wesayc is an inflection point of f if f changes concavity at C 5 (Mean Value Theorem - 3 Points). Let f be a function satisfying 1. f is continuous on [a, b], and 2. f is differentiable on (a, b). oaf'lx f Caco of lx f Cxko negative. positive Then there exists a c in (a, b) suchthat. f 0 (c) = fast food b a
Problems 6 (15 Points). Find dy/dx: cos(xy) =1+tany sedly date sin Gyllyt x 1 xsincx ysinxy g If seecgladytxsinlxgldd Y I ffsedcyltxs.in Gg y sinkly iiiy EXAM 2 3 7 (20 Points). Astreetlightismountedatthetopofa15-ft-tallpole. Aperson6fttall walks away from the pole with a speed of 5 ft/s along a straight path. How fast is the tip of their shadow moving when they are 40 ft from the pole? (Hint: The length of the shadow is measured from the person to the tip of the shadow; the rate at which the tip of the shadow is moving is measured from the pole to the tip of the shadow.) IS X XitXz 4 We are given 1 5 and we want to find 1 have 6 I Gx 15 2 15 X Xi tox tox 42 X 6 15 9x tox x tox 53X Differentiating both sides yields dft E.EE EG YET By similar triangles we
4 EXAM 2 8 (15 Points). Find the value or values of c that satisfy the conclusion of the Mean Value Theorem for the function f(x) =x 3 2x 2 4x +2on[ 2, 2]. f 2 8 264 412 t2 8 8 8 2 6 fl 2 8 264 4C2 12 8 81812 6 6 fkz.fii y Ei 9 (15 Points). Find the absolute maximum and minimum values of f(x) =2x 3 3x 2 12x+1 on [ 2, 3]. oo f'cx1 3x2 4x 4e sx f4hztyyf 4t 5 6 4 ft 4t 6 4 41 4 21 Elite Either 34121 2 which is not in C321 or c 5Ctz f 6 6 2 6 12 61 2 X 2 6 X 2 Xt 1 O X 1 or X 2 off 2 2Es 314 12127 1 16 12124 1 161121 1 3 fl l 2Cl 3G 1211111 2 3 121 1 8 f 2 2 8 3 4 12121 1 16 12 24 t I 4 24 1 20 1 19 f 3 267 3G 1213 H 27 2 1 36 1 27 35 8 Absolutemaxic Absolute min G 19
EXAM 2 5 10 (20 Points). Sketch the curve (a) State the domain of f. 7 f(x) = x2 x 1 (b) Find the intercepts and express them as an (x, y) pair. Write NONE if there are none. x-intercepts: y-intercept: 0,01 0,01 (c) Is the function even, odd, or neither? What type of symmetry does the function have? ffxt.fi Yn tflxhflxl neither nosynmetryq (d) Find the asymptotes. Write NONE if there are none. Horizontal: Oblique: Vertical: NONE y xtl XT fins 9 m i 1 Eat x fig no x a a ix xtit Ein D i t I H FEE o
6 EXAM 2 (e) Find the intervals where the function is increasing and decreasing. Write NONE if not applicable. Increasing: Decreasing: f4xi t 2xlx y X2ZX or com 2,01 C o o and za o l U 1,2 since I is not inthe domain of f x4 2x2 (f) State the local maximum and local minimum value(s). Write NONE if not applicable. Local maximum value(s): Local minimum value(s): 0,0 2,41
EXAM 2 7 (g) Find the intervals on which the function is concave up and concave down. State the inflection points. Write NONE if not applicable. Concave Up: Concave Down: Inflection Points: 1,0 C x 11 NONE f'm 12 211 42 6224611 1141 X 112 F It zfx.nl x it xcx2d 2fx 2x tl x t2 x t X l't 1 113 X 1 53 FO (h) Use your answers to Parts (a)-(g) to sketch the curve. Be sure that your graph is labeled and neat. Messy/incoherent graphs will receive zero points. XA t yal l l viol il l I b't t't