version 1.0β, 2008-04-11 06:38Name: ID#:Midterm IIMath 1aIntroduction to CalculusApril 11, 2008Rules:• This is a one-hour exam.• Calculators are not allowed.• Unless otherwise stated, show all ofyour work. Full credit may not begiven for an answer alone.• You may use the backs of the pages orthe extra pages for scratch work. Donot unstaple or remove pages as they canbe lost in the grading process.• Please do not put your name on anypage besides the first page. If you like,you may put your ID number on thetop of each page you write on.• Please, please, please don’t cheat.Hints:• Read the entire exam to scan for ob-vious typos or questions you mighthave.• Budget your time so that you don’trun out.• Problems may stretch across severalpages.• Relax and do well!Good luck!version 1.0β, 2008-04-11 06:38Problem Possible PointsNumber Points Earned1 102 103 104 105 106 10Total 60version 1.0β, 2008-04-11 06:381 Math 1a Midterm II April 11, 2008 11. (10 Points) For each of the following functions of x, circle the derivative. No justificationis necessary.(i) arcsin(x)Note. Our book writes this function as sin−1(x).(A) − csc x cot x (B)11 + x2(C) sec x (D)1√1 − x2(ii) ln(x2+ 1)(A)x2+ 12x(B)2xx2+ 1(C)12x(D)1x2+ 1(iii) 2x2(A) 2x2(2x) (B) (ln 2)2x2(C) (ln 2)x2x2+1(D) x22x2−1(iv) xsin x(A) ln x · cos x · xsin x(B) sin x · xsin x−1(C) ln x ·cos x · xsin x+ sin x · xsin x−1(D) 1sin x+ xcos x(v) log3π(A)ln 3ln π(B)ln πln 3(C) 0 (D)1π ln 3/ 10–1–version 1.0β, 2008-04-11 06:382 Math 1a Midterm II April 11, 2008 22. (10 Points) Consider the curve9x2+ 16y2= 5625(a) (5 points) Find the slope of the line tangent to the curve at (7, 18).(b) (3 points) Find the y-intercept of the line tangent to the curve at (7, 18).(c) (2 points) Suppose x is increased to 7.32. The corresponding y value on the curve is:(A) greater than 18.07(B) less than 18.07 but greater than 18(C) less than 18 but greater than 17.93(D) less than 17.93(No justification necessary. Just circle the correct answer)/ 10–2–version 1.0β, 2008-04-11 06:383 Math 1a Midterm II April 11, 2008 33. (10 Points) For each of the following statements, indicate whether it is true or false. Nojustification is necessary.(i) T F The absolute minimum value of f (x) = 1 − x2on [−1, 2] is −3.(ii) T F If f has a local minimum at c and f is differentiable at c, then f00(c) > 0.(iii) T F An inflection point is always a critical point.(iv) T F If f has a relative maximum at at c and f is differentiable at c, then f0(c) = 0.(v) T F If f0(c) = 0, then f has a relative extremum at c.(vi) T F If c is a critical point of f , then f0(c) = 0.(vii) T F If f00(x) > 0 for all x and f0(c) = 0, then c is the global minimum of f(viii) T F If f0(x) > 0 on (a, b), then f is increasing on (a, b).(ix) T F If f00(x) > 0 on (a, b), then f is concave up on (a, b).(x) T F If f0(c) = 0 and f0(c) > 0, then c is a relative minimum./ 10–3–version 1.0β, 2008-04-11 06:384 Math 1a Midterm II April 11, 2008 44. (10 Points) Find the following limits. Explain your reasoning.Two of these are three points and two are two points.(i) limx→∞h12x(x −1)i2x4(ii) limx→01 −cos xx2(iii) limx→−π2x +π2sin x + 1(iv) limx→0xex−1/ 10–4–version 1.0β, 2008-04-11 06:385 Math 1a Midterm II April 11, 2008 55. (10 Points) Romeo and Juliet are on opposite sides of a circular pond of radius 1 mi.To reach Juliet, Romeo can run around the pond at a speed of 5mi/hr, or he can swim at aspeed of 2mi/hr. He decides the best thing to do is run part of the way around then swimthe rest of the way in a straight line, so as to minimize the total time it takes to reach hislove.Set up a calculus problem for Romeo to do. Draw a picture and label it with appropriatenotation. Express the objective function as a function of a single variable and give thedomain of that function. You do not have to solve the actual problem—we’ll leave that forRomeo!/ 10–5–version 1.0β, 2008-04-11 06:386 Math 1a Midterm II April 11, 2008 66. (10 Points) We’re starting a business selling political buttons. The cost of making xbuttons is given by the functionC(x) = x3+ 2x2+ 100x + 2200where C(x) is measured in dollars and x is the quantity in thousands of buttons.How many buttons should we make to minimize average cost? What is that minimumaverage cost?/ 10–6–version 1.0β, 2008-04-11 06:38(This page intentionally left blank. You can use it for scratch
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