Part I consists of 14 multiple choice questions (worth 5 points each) and 5 true/false question (worth 1 point each), for a total of 75 points.

Transcription

1 Math 131 Exam 1 Solutions Part I consists of 14 multiple choice questions (orth 5 points each) and 5 true/false question (orth 1 point each), for a total of 75 points. 1. The folloing table gives the number of honeybees ÐLÑ in a colony at time eeks. L! * (& &! % & & & '!& What as the average rate of change of the honeybee population (honeybees/eek) from! to? ( Round your anser to the nearest tenth. ) A) 7.1 ) 5.3 C) 4.3 D) 19.5 E) 19.7 F) 4.0 G) 4.6 H) 1.5 I) 1.9 J) 15.4 LÐÑLÐ!Ñ &! honeybees! eeks %Þ! honeybees/eek. The anser is negative because the population has a net decrease over this time period.. ( Still using the table from Problem 1) y averaging to estimates (one too small, the other too large), hat is your best estimate from this data for the rate of change of the honeybee population (honeybees/eek) hen? ( Round your anser to the nearest tenth.) A) 7.1 ) 5.3 C) 4.3 D) 19.5 E) 19.7 F) 4.0 G) %Þ' H) 1.5 I) 1.9 J) 15.4 LÐÑLÐÑ LÐÑLÐÑ Since the population is decreasing from! to À and are the estimates from the nearest data to : one too large, the other too small. LÐÑLÐÑ LÐÑLÐÑ Averaging them gives our best estimate of Ò Ó % Þ& honeybees/eek. (Which estimate is too small and hich one is too large?)

2 È 3. A point moves according to the parametric equations C(Þ Exactly three of the folloing five statements are true. Which three are true?!ÿÿþ i) the point is moving along part of a parabola ii) the point moves left to right along its path iii) the point is moving along part of a circle ÈÐÑ iv) the equations C(ÐÑ describe exactly the same path È? v) the equations? C( % Þ describe exactly the same path.!ÿÿ!ÿ?ÿ A) iii, iv, v ) ii, iv, v C) ii, iii, v D) ii, iii, iv E) i, iv, v F) i, iii, v G) i, iii, iv H) i, ii, v I) i, ii, iv J) i, ii, iii i) T: Since, e have C(Ð Ñ%, hich is a parabola ii) T: As increases, so does so the point is moving rightard iii) F: (See i) ) iv) F: these equations describe a curve ending at Ð È( ß Ñ; the given equations describe a curve ending at Ðß )ÑÞ v) T: these equations are just the result of substituting? in the given equations and adjusting the range of the parameter.

3 4. The folloing graph shos the position =0ÐÑof a car moving along a straight road at time. The points A,, C, D on the graph represent the position of the car at certain times. Which of the folloing statements are true? i) The car as moving faster hen it as at the position corresponding to than hen it as at the position corresponding to A. ii) The car as moving backards for Ÿ Ÿ Þ& iii) The graph indicates that there ere exactly 5 times (not necessarily corresponding to A,,C, or D Ñ hen the car had velocity 0 iv) The car as speeding up as it moved from the position corresponding to A to the position corresponding to. v) During the time hen the car moved from the position corresponding to A to the position corresponding to D, the average velocity as approximately 0. A) iii, iv, v ) ii, iv, v C) ii, iii, v D) ii, iii, iv E) i, iv, v F) i, iii, v G) i, iii, iv H) i, ii, v I) i, ii, iv J) i, ii, iii

4 Recall that the slope of the tangent line at a point the velocity at that time i) F: The tangent line has a larger slope at A, and the slope represents the car's velocity. ii) T: For Ÿ Ÿ Þ&, the slope of the tangent velocity is negative iii) T: There are exactly 5 places on the graph (tops of peaks or bottoms of pits ) here the tangent line is horizontal (that is, has slope!). iv) F: It's sloing don because the slope of the tangent line (=velocity) is decreasing beteen A and v) T: the average velocity is the slope of the secant line through A and D and that line is approximately horizontal., & 5. Let 0ÐÑ. What value of, makes 0 continuous?,, A), 1 ), C), 3 D), 4 E),5 F), % G), H), I), J),! For 0ÐÑ to exist, e need 0ÐÑ, &, 0ÐÑ, so, Þ If,, then 0ÐÑ(0ÐÑßso 0 is then continuous at. 6. Suppose the position of a point (in cm) moving along a straight line is =0ÐÑ40 at time (sec). What is its instantaneous velocity hen? A) cm/sec ) cm/sec C) 3 cm/sec D) cm/sec E) cm/sec F) cm/sec G) cm/sec H) cm/sec I) cm/sec J) cm/sec 0ÐÑ0ÐÑ Ð%! ÑÐ%! Ñ Ð Ñ Ð Þ

5 7. The Intermediate Value Theorem tells us that the equation /! has a solution - in the interval Ò!ß ÓÞ y repeatedly bisecting intervals, e can find an interval as short as e like that contains -. What is the smallest number of bisections needed to enclose - in an interval of length Ÿ 0.01? A) 1 ) C) 3 D) 4 E) 5 F) 6 G) 7 H) 8 I) 9 J) 10 st After the 1 bisection e have subintervals, Ò!ß Þ&Ó and ÒÞ&ß Ó, each of length and one of hich contains -. nd After the bisection, e have 4 intervals each of length and one of hich contains th 8 After the 8 bisection e have subintervals, each of length and one of hich 8 contains -. We ant to continue until the length of the subintervals Ÿ!Þ!. This happens for 8 the first time hen 8*. ( The easiest ay to get 9 is just by trial/error using a calculator; hoever, you could also solve the inequality using logarithms.) % + % ( 8. The function D0ÐÑ has to horizontal asymptotes: D. What is +? È A) '% ) C) 9 D) * E) È F) È G) 0 H) ' I) J) % Ð% ÑÎ % Ä_ È + % ( Ä_ Ð È + % ( Ñ Î Ä_ É % % ( Ä_ É+ % + È and ' + % ( % Ð% ÑÎ % Ä_ È + % ( Ä_ Ð È + % (ÑÎ Ä_ É % % Ä_ % ( É+ È+ % È+ Þ Therefore, so + '%Þ '% + % (

6 9. Which set of parametric equations has the graph shon belo? sin sin A)!ŸŸ 1 ) C!ŸŸ1 C C)!ŸŸ D) 1!ŸŸ1 C sin C sin 3 cos 3 cos E)!ŸŸ 1 F) sin C C sin!ÿÿ1 sin 3 cos G)!ŸŸ 1 H) 3cos C C 3cos!ŸŸ 1 I)!ŸŸ 1 J)!ŸŸ 1 C C In the graph, ranges back and forth beteen. That einates all pairs of equations except A), ), G). Also in the graph, C is alays 0. That einates G) 1 1 For!ŸŸ1ß sin only hen (so that Ñ. ut in the picture occurs three times. That einates ), leaving only A)

7 Ð ÑÐ % %Ñ Ä For hat value of + is 1? A) +! ) + 1 C) + D) + 3 E) +4 F) + 5 G) + 4 H) + 3 I) + J) + 1 Ð ÑÐ % %Ñ Ð ÑÐ Ñ Ä+ + Ä+ +. Since the denominator Ä 0 as Ä +, the it can possibly exist only if the numerator Ä 0 also as Ä +Þ That means e need either + or + Þ Ð ÑÐ Ñ If +, e get!þ Ä Ð ÑÐ Ñ Ä If + ße get hich is hat e ant. 11. The size of a population of rabbits on a small island at time years is TÐÑ &ß!!!/!!/ *!!. As time passes, the population eventually levels off toard a size called the carrying capacity of the island (for rabbits). What is the carrying capacity for this island? A) 100 ) 50 C) 350 D) 900 E) 1000 F) ) 500 G) 3500 H) 9000 I) J) &ß!!!/ Ð&ß!!!/ ÑÎ/ &!!! Ä_!!/ *!! Ä_ Ð!!/ *!!ÑÎ/ Ä_!! &! *!! / 1. The function 0ÐÑ Ð ÑÐ Ñ Ð Ñ Ð Ñ has ho many vertical asymptotes? A) 0 ) 1 C) D) 3 E) 4 ( Only 5 choices are intended in this problem. ) The denominator Ä! as Ä!ßßÄ. There are no other points here 0ÐÑmight Ä _. So e check hat happens as Ä!ßßÄ. We can rearrange the function as ÐÑ Ð Ñ Ð Ñ 0ÐÑ Ð ÑÐ Ñ Ð Ñ Ð Ñ Ð ÑÐ Ñ Ð ÑÐ Ñ. In this form, it is clear that the function blos up near! and Ð Ñ, so there are exactly vertical asymptotes.

8 13. For the function C 0ÐÑ / /, it turns out that 0 ÐÑ Ð/ / Ñ. What is the equation of the tangent line to the graph at the point here!? A) C ) C C) C D) C% E) C% F) C% G) CÐ/Ñ% H) CÐ%/Ñ I) C% J) C!! 0 Ð!Ñ Ð/ / Ñ ÐÑ % gives the slope of the tangent line.!! When!, C 0 Ð!Ñ / /!Þ The line through Ð!ß!Ñ ith slope % has equation ÐC!Ñ %Ð!Ñ or C %Þ sin 5ÐÑ 14. Let 0ÐÑ Þ For hat value of 5 ill 0ÐÑ exist? 5 55 A) 50 ) 5 C) 5 D) 5 E) 5 % F) 5 & G) 5% H) 5 I) 5 J) 5 sin 5ÐÑ 5 sin 5ÐÑ 5 sin? 5ÐÑ?Ä!? (here sin? 5 5Þ?Ä!? Ð Ñ 0ÐÑ 5Ð ÑÐ Ñ 5Ð Ñ 5Þ 0ÐÑ? Ñ To make the it exist, e need to have 5 5ß so 5 Þ 0ÐÑ 0ÐÑ, that is, Questions 15)-19) are true/false questions 15. The Intermediate Value Theorem guarantees that the equation %%! has a root in the interval Ò!ß %Ó. A) True ) False If 0ÐÑ %%, then 0Ð!Ñ % and 0Ð%Ñ %Þ Therefore! is not a number beteen 0Ð!Ñand 0Ð%Ñ so the Intermediate Value Theorem doesn't apply. (In fact, using the quadratic formula, the roots are % È '' È, both of hich are outside the interval [0,4].

9 tan 16. C has a vertical asymptote at!. A) True ) False tan sin sin Ä! Ä! cos Ä! cos Þ % 17. gives the slope of the tangent line to C È at the point Ð%ß Ñ. È A) True ) False 0 Ð%Ñ the slope of the tangent line at Ð%ß Ñ È% È% È% 0Ð%Ñ0Ð%Ñ 18. The Squeeze Theorem tells us that sinð Ñ _Þ A) True ) False Ä_ It's true that e can rite Ÿsin Ÿ and conclude that Ÿsin Ÿ. ut as Ä_, this just says that sin is trapped beteen to quantities one of hich Ä _ and the other of hich Ä _. This doesn't let us conclude anything about hat happens to sin Þ sin sin? Ä_?Ä!? In fact, sin Ð here? Ñ Þ Ä_ 19. Suppose =@ÐÑis velocity of a point moving along a straight line. For a approximates the point's acceleration at time. A) True ) False The acceleration + is the rate of change of the velocity ith respect to At time, This means that gets close @ ÐÑ for small values of.

10 Name ID Number Please put your name and ID number above and on each folloing sheet of Part II, in case the sheets get separated during the grading process. Part II: (5 points) In each problem, clearly sho your solution in the space provided. Sho your solution does not simply mean sho your scratch ork you should cross out any scratch ork that turned out to be rong or irrelevant and, here appropriate, present a readable, orderly sequence of steps shoing ho you got the anser. Generally, a correct anser ithout supporting ork may not receive full credit. 0. Find each it (if the it does not exist, explain hy). a) ( Ä ' ( Ð ÑÐ Ñ Ð Ñ Ð'Ñ ) Ä ' Ä ÐÑÐÑ Ä ÐÑ & & Ð Ñ sin Ð Ñ sin Ð Ñ ÐÑ b) Ð Ñ sin Ð Ñ sin Ð Ñ Ð ÑÐ Ñ sin Ð Ñ sin Ð Ñ ÐÑ ÐÑ Ð Ñ sinð Ñ sin ÐÑ sin Ð Ñ sin Ð Ñ ÐÑÐÑ ÐÑ sin Ð Ñ sin Ð Ñ sin? ÐÑ?Ä!? sin Ð Ñ Þ sin Ð Ñ sin Ð Ñ Ð Ñ ut (here? Ñ Þ Similarly Therefore ÐÑÐÑ 'Þ c) È * Ä! È * È * È * Ä! Ä! È * ** Ä! Ð È * Ñ Ä! È * ' ÐSee Text, p. 114) Þ

11 1. a) Let 0ÐÑ. Write, in it form, the definition of 0 ÐÑ. ( Substitute into the specific function 0 given here. Do not actually ork out the value of the it.) 0ÐÑ0ÐÑ ÐÑ ÐÑÐ Ñ 0 ÐÑ OR 0ÐÑ0ÐÑ Ä 0 ÐÑ Ä ÐOf course, on simplification, both its end up having the same valueþñ b) Suppose C1ÐÑ È i) Use the it definition to find 1Ð'Ñ 1Ð' Ñ 1Ð'Ñ È* È* È* È* È* ** Ð È* Ñ È* ' 1Ð'Ñ ii) What is the equation of the tangent line to C1ÐÑ here '? 1Ð'Ñ, so e ant the tangent line through Ð'ß ÑÞ Its slope is 1 Ð'Ñ ' Þ Therefore the tangent line has equation ÐCÑ Ð'Ñ or C Þ ' ' iii) If C represents the value () of an investment after years, hat are the units of 1Ð'Ñ? If C is in and is in years, then 1 Ð'Ñ has units /year (the rate of change of the value of the investment ith respect to time).