MATH Fall 08. y f(x) Review Problems for the Midterm Examination Covers [1.1, 4.3] in Stewart

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1 MATH Fall 08 Review Problems for te Midterm Eamination Covers [1.1, 4.3] in Stewart 1. (a) Use te definition of te derivative to find f (3) wen f() = π 1 2. (b) Find an equation of te tangent line at (3, π 5 ). 2. Consider te function f() = Find an integer n suc tat f(n) < 0 and f(n + 1) > 0. Sow tat tere is a real number c suc tat n < c < n + 1 and f(c) = 0 (name te teorem used). 3. Te derivative of = f() at = 3 is defined as (A) lim f(3 + ) f(3) (D) lim f( + 3) f(3) (B) lim 0 f(3 + ) f(3) (E) lim f( + 3) f(3) (C) lim f(3 + ) f(3) 4. Consider te function f defined b te following grap. 3 2 f() 1!4!3!2! !1 (a) f is not continuous at =. (b) f does not ave a derivative at =. (c) lim 2 f() =. (d) lim 1 f() =. (e) f( 2) =. 5. Compute eac of te following limits eactl (sow our work) or state DNE: (a) lim (b) lim + 5 (c) lim

2 π (d) lim (e) lim 1 ( 1) (g) lim () lim 2 ( 2) 2 π (f) lim t 0 (1 + t) t (i) lim 0 1 cos sin (j) lim [[]] (Hint: Sketc te grap.) 3 cos 1 (k) lim 1 cos (l) lim 17(sin ) 1 cos (m) lim 0 π(sin )(1 + cos ) (n) lim 0 1 cos (2) 17 2 (o) lim 2 (1 + ) ( 2 3/2 ) (p) lim 10 [[2 ]] (q) lim π e sin 1 π (r) lim θ π 3 (u) lim 0 2 cos θ 1 1 θ π. (s) lim ln( (v) lim Find te interval(s) on wic te function is continuous. (a) f() = 2 4 (b) g() = Solve te equation for. ) (t) lim 0 tan 2 + sin sin 2 (c) = ln 1 (d) (t) = 16 t 2 (a) e 2+1 = 3 (b) e 2 +1 = e (c) ln( + 1) = 2 (d) ln + ln( 1) = ln 2 cos sin 8. lim = (A) 1 (B) 1 (C) 2 (D) (E) (F) DNE ( ) lim ( 1) sin = (A) 0 (B) 1 (C) 2 (D) (E) (F) DNE 1 1 { 2 3 > Let f() =. Te function is not defined at = 1. Can a value of f(1) be 1 2 < 1 assigned to make f continuous at = 1? If so, give te value of f(1). If not eplain w not. 11. Let f() = { a 2 > 1. For wic value of a will f be continuous at = 1? 12. Let f() = { c c < 1. For wic value of c will f be continuous at = 1? 13. Let f be a function defined on [0, 100] suc tat lim f() = lim f() = π. Wic of te following must be true? 2

3 (a) f(17) = π (b) f(π) = 17 (c) lim f() = π (d) f is continuous at = (e) f is differentiable at = Te function f() satisfies f(4) = 489 and f (4) = 378. Wic of te following values is te best linear approimation of f(4.13)? (a) (b) (c) (d) (e) Consider te function f defined b te following grap. Put DNE (does not eist) if applicable. 3 f() 2 1!4!3!2! !1 f( + ) f() (a) f is not continuous at =. (b) lim does not eist at =. (c) lim f() =, (d) lim f() =. (e) f(2) = Use te definition of te derivative to find te following: (write carefull and include all te steps.) (a) f () wen f() = 1. (b) g () wen g() = + 1. (c) k () wen k() = At = 0, te function given b f() = (a) undefined. (b) continuous but not differentiable. (c) differentiable but not continuous. (d) neiter continuous nor differentiable. (e) bot continuous and differentiable. 18. If lim f() = π and lim f() = 3.14, ten (a) lim 2 f() = π and f is continuous at = 2. { sin 0 > 0 is 3

4 (b) lim 2 f() = π and f is not continuous at = 2. (c) lim 2 f() does not eist and f is continuous at = 2. (d) lim 2 f() does not eist and f is not continuous at = 2. (e) None of te above. 19. lim 0 tan sin 3 = (a) 1 2 (b) 1 (c) 2 (d) (e) DNE 20. If f(3) = 0, f () 2 for 2 < < 4, and f () 2 for 2 < < 4, ten (a) f is increasing and concave upward at = 3. (b) f is decreasing and concave upward at = 3. (c) f is increasing and concave downward at = 3. (d) f is decreasing and concave downward at = 3. (e) None of te above. 21. Te following data is known about te function f and g. Find te value of: (a) d [f(t)g(t)] at t = 3. dt (b) d ( ) g(t) at t = 5. dt t 2 (c) d dt (et f(t)) at t = 0. t f(t) f (t) g(t) g (t) (d) d ((f(t) + g(t)) at t = 1. dt 22. Te derivative of f() = cos( 2 ) at = 0 is given b te limit (a) lim cos ( 2 ) cos () (b) lim cos ( 2 ) sin ( 2 ) 4

5 (c) lim cos ( 2 ) 1 (d) lim cos () 1 (e) None of te above 23 (a) Sketc te curve represented b te parametric equations = ln t, = t 2 for 1 t 3. Indicate wit an arrow te direction in wic te curve is being traced as t increases. (b) Eliminate te parameter to find as a function of. 24. lim (17 π)( 17) (π 17)(17 1) = (a) 17 (b) π (c) 17 π (d) π 17 (e) 1 π. 25. Let g be a function suc tat g() is defined for all π. Also, suppose tat g as a continuous second derivative for π. Also, 0 = g(0) = g(4), g( 17) = 3.2, g( e) = 2.1, lim g() = 2.3, lim g() =, lim g() =, lim g() = +. + π + π Te following information is also known about g and g. < < < e e < < 0 0 < < π π < g () g () + + Find (no justification needed): (a) Te coordinates of all local maima. (b) Te coordinates of all local minima. (c) Te coordinates of all inflection points. (d) Te equations of all orizontal asmptotes. (e) Te equations of all vertical asmptotes. (f) Carefull sketc a grap of g below: 3 2 1!6!5!4!3!2! !1 26. Te curve of function f is sown in te figure. On te same set of aes, sketc te grap of te derivative f. 5

6 f() 27. Te curve of te derivative function f () is sown in te figure. Suppose f(0) = 0. On te same set of aes, sketc te grap of function f(). f () 28. (a) Sketc te curve given b te parametric equations = sin t 2, = 2 cos t 2 for 0 t π. Indicate wit an arrow te direction of te curve as t increases. (b) Eliminate te parameter t to find an equation in and. 29. Consider te curve defined b = 80. Find d and an equation for te line d tangent to te curve at te point (4, 2) 30. Let g and be differentiable functions suc tat g(1) = (1) = 1, g (1) = 3, (1) = 3, g (2) = 4, and (2) = 5. If m() = g(()), ten m (1) = (A) 9 (B) 4 (C) 0 (D) 12 (E) If f() = 2 and g() = ln, find te functions f g, f f, g f, g g and teir domains. 32. If f is a function of suc tat f () > 0 for all and f () < 0 for all, wic of te following could be a part of te grap of = f()? 6

7 (A) (B) (C) (D) (E) 33. A particle moves along a line so tat at an time t its position is given b s(t) = 2πt + 2 cos(2πt). (a) Find te velocit v(t) at time t. (b) Find te acceleration a(t) at time t. (c) Find te values of t, 0 t 1, for wic te particle is at rest. 34. Let f() = 4 2. For 0 < w < 4, let A(w) be te area of te triangle formed b te coordinate aes and line tangent to te grap of f at te point R = (w, 4 w 2 ). (See te figure) (0,4) R (w,4!w ) 2 f() 0 (2,0) (a) Find te equation of te line tangent to te grap of f at te point R. (b) Find an epression for A(w). (c) Find te rate of cange of te area of te triangle wit respect to w. 35. If f() = 3 + 1, find (a) f(2), (b) f 1 (2). 36. Consider te equation = sin. Find (a) d ( d 4 te point π 2, π ). 4 and (b) an equation of te tangent line troug 7

8 37. Given tat v() = u(w()), find te si missing values in te below table: w() w () u() u () v() v () ? 2 3 0? 6?? ?? Consider te equation = tan. Find d ( d 4 π, π ). 4 and an equation of te normal line to te grap at 39. Calculate te slope of te tangent line to te grap of cos( ) = sin at te point (0, π 2 ). 40. Consider te equation sin = cos. Find d2 in terms of and. d2 41. A cclist moving at 14 km/r first rides nort 8 km and ten turns east. How fast is er distance from er starting point canging after one our? 42. A ladder, 20 feet long, leans flus against a wall, i.e., te ladder is vertical. Te foot of te ladder is pulled awa from te wall at a constant velocit of 1 ft/sec. How fast is te top of te 2 ladder moving downward wen te foot of te ladder is 9 feet from te wall? 43. Sketc a grap of a single function tat as all te following properties. (a) () is defined and continuous for all 0 (b) () eists for all 0 (c) lim () = 0+, lim () =, lim () =, lim () = (d) ( 2) = 2, ( 2) = 0, ( 2) < 0, (1) < 0, () > 0 for all > Sketc te curve given b te parametric equations = t 2, = t 3 for 0 t 3. Indicate wit an arrow te direction of te curve as t increases. Eliminate te parameter t to find a Cartesian equation wit and. 45. Te potential energ V acting on a particle at a distance q = 3 is known to be V (3) = 2.5. If te derivative of V wit respect to te variable q at q = 3 is V (3) = 0.6, use linear approimation to determine te approimate value of te potential at q = If f and g are te functions wose graps are sown, let P () = f()g(), Q() = f()/g(), and C() = f(g()). Find (a) P (2) (b) Q (2) (c) C (2). 8

9 g f 47. Find te point(s) on te curve = π 2 at wic te tangent line is orizontal. 48. Find te point(s) on te curve at wic te tangent line is parallel to =. = 2 tan, π 2 < < π Let g() = f() 3. Find g ( 5) if f ( 5) = 14 and f( 5) = (a) Find an equation of te tangent line to te curve = e tat is parallel to te line 4 = 1. (b) Find an equation of te tangent line to te curve = e tat passes troug te origin. 51. Find g () if it is known tat d d (g(17) + 2 ) = A car is travelling at nigt along a igwa saped like a parabola wit its verte at te origin. Te car starts at a point 100 m west and 100 m nort of te origin and travels towards te origin. Tere is a statue located 100 m east and 50 m nort of te origin (te statue is not on te igwa). At wat point on te igwa will te car s eadligts illuminate te statue? 53. If = ln d, ten d = (a) 1 (b) 1 ln 1 (c) If = (2), for > 0, ten d d = 55. (d) 1 ln 2 (e) 1 + ln 2 (A) 2 (2 1) (B) 2 (2 +1) ln (C) + 2 ln (D) (2 +1) (1 + 2 ln ) (E) 3 2 d d (ln ) = (A) ln (B) (ln ) (C) 2 (D) (ln )(ln 1 ) (E) 2(ln )( ln 1 ) 56. Two cars start moving from te same point at te same time. One car travels nort at 30 mp 9

10 and te oter car travels east at 40 mp. How fast is te distance between te cars increasing two ours later? 57. A man starts walking nort at 4 ft/sec from a point P. Five minutes later, a woman starts walking sout at 5 ft/sec from a point 500 ft due east of P. At wat rate are te walkers moving apart 15 minutes after te woman starts walking? 58. Find te point(s) on te curve = 2 2 at wic te tangent line passes troug te point (1, 5) 59. A runner runs around a circular track of radius of 100 meters at a constant speed of 7 m/sec. Te coac is standing at a point 200 meters from te center of te track. Te runner starts running at te point on te track nearest te coac. Determine ow fast te distance between te coac and te runner is canging wen te distance between tem is 200 meters. (Hints: Arc lengt formula: s = rθ, Law of Cosines: a 2 = b 2 + c 2 2bc cos θ.) 60. A swimming pool is 15 feet wide, 40 feet long, 3 feet deep at one end, and 10 feet deep at te oter end (see te figure). Te pool is being filled wit water at te rate of 10 ft 3 /min. How fast te dept of te water is rising wen te water is 4 feet deep? For wic of te following functions does te propert d3 d 3 = d d old? I. = e II. = e III. = sin (A) I onl (B) II onl (c) III onl (D) I and II (E) II and III 62. Consider te curve described b = e. (a) Find d d te tangent line to te curve at = 0, = 1. in terms of,. (b) Find an equation of 63. Te cost, in dollars, of producing units of a certain commodit is C() = (a) Find te marginal cost function. 10

11 (b) Find C (100) and eplain its meaning. (c) Compare C (100) wit te cost of producing te 101st unit. 64. Te normal line to te curve represented b te equation = at te point ( 2, 4) also intersects te curve at anoter point (, ). Find te point (, ). Sow our work! 65. A particle moves along a line so tat at an time t its position is given b were t is measured in seconds and s in meters. (a) Find te velocit v(t) at time t. (b) Find te acceleration a(t) at time t. (c) Wen is te particle at rest? s(t) = 2πt + 2 cos(2πt), 0 t 1, (d) At t = 1 sec, wat is te position of te particle? Is it moving forward (in te positive 6 direction)? Is it speeding up or slowing down? 66. Consider a differentiable function wit te properties tat (17) = 22 and (17) = 2. Ten a linear approimation of (18.5) is (a) 17 (b) 22 (c) 28 (d) 23.5 (e) Te conical tank sown as a radius of 160 cm and a eigt of 800 cm. It is partiall filled wit water wit te eigt of te water denoted b. 160 m 800 m (a) Find te volume of te water as a function of onl. (b) If water is leaking out of a ole at te verte of te cone, wat is te rate of cange of te volume wit respect to te eigt? 68. Te large stucco teepee (nort of Lawrence) needs painting. Te eigt is 20 ft and te diameter is 40 ft. A coat of paint is.02 in tick. Use differentials to estimate te number of gallons required for one coat of paint. [Hint: 1 ft 3 = 7.48 gallons.] 11

12 69. A cubic block of ice is melting in suc a wa tat eac side is decreasing at te same rate. Use differentials to approimate te cange in volume from wen te side lengt canges from 5 cm to 4.9 cm. 70. Suppose f is a twice differentiable increasing function wit eactl two points of inflection on [a, b]. Wic of te following is a possible grap of f? a b a b a b (A) (B) (C) a b a b (D) (E) 71. If a population of bacteria starts wit 1000 bacteria and doubles ever our, ten te number of bacteria after t ours is given b N(t) =1,000 2 t. (a) Wen will te population reac 150,000? (b) Wat will be te rate of growt of te bacteria population at tat time? 72. Based on our observation onl, decide weter eac of te following statements is true or false for te function = f () graped below.!6!5!4!3!2! T F (a) f is continuous at te point = 4. T F (b) f is differentiable at te point = 4. T F (c) f is continuous at te point = 1. T F (d) f is differentiable at te point = 1. 12

13 T F (e) f is continuous at te point = 2. T F (f) f is differentiable at te point = A eav weigt is trown from a rooftop 160 feet ig so tat at t seconds, its eigt above te ground is s (t) = 16t t feet. Decide weter eac of te following statements is true or false. T F (a) Te weigt its te ground after 2 seconds. T F (b) Wen it its te ground, its velocit is positive. T F (c) Its acceleration is alwas positive. T F (d) Its acceleration is alwas negative. T F (e) Its acceleration is a constant. 74. A function f defined on te set of all real numbers as te following properties. If < 2 or 3 <, ten f () > 0. If 2 < < 3, ten f () < 0. f (3) = 0 but f (2) is undefined. If < 0, ten f () < 0. If 0 < < 2 or > 2, ten f () > 0. Decide weter eac of te following statements is true or false. T F (a) f is increasing on te interval (2, 3). T F (b) f is increasing on te interval (, 2). T F (c) f is concave down on te interval (2, ). T F (d) f is concave down on te interval (, 0). T F (e) f as a relative maimum at = 3. T F (f) f as a relative maimum at = 2. T F (g) f as a relative maimum at = A paper cup as te sape of a cone wit eigt 10 cm and radius 3 cm (at te top). If water is poured into te cup at a rate of 2 cm 3 /s, ow fast is te water level rising wen te water is 5 cm deep? 76. A balloon is rising at a constant speed of 5 ft/s. A bo is ccling along a straigt road at a speed of 15 ft/s. Wen e passes under te balloon it is 45 ft above im. How fast is te distance between te bo and te balloon increasing 3 s later? 77. A street ligt is mounted at te top of a 15 ft pole. A man walks awa from te pole at te speed of 5 ft/s along a straigt pat. How fast is te tip of is sadow moving wen e is 40 ft from te pole? 78. Find te absolute maimal and minimal value of te function f() = Find te critical numbers of te function f(θ) = 2 cos(θ) + sin 2 (θ) 13

14 80. Find te absolute maimal and minimal value of te function f() = e 2 / Find te local and absolute etreme values of te function f() = For te function f() = 2 2 1, determine te vertical and orizontal asmptotes, te intervals of increase/decrease and te local maimums and minimums. 83. Determine te te intervals of increase/decrease, te local maimums and minimums and te intervals of concavit and te inflection points for f() =