MATH 151, SPRING 2011COMMON EXAM III - VERSION ALAST NAME, First name (print):INSTRUCTOR:SECTION NUMBER:UIN:SEAT NUMBER:DIRECTIONS:1. The use of a calculator, laptop, or computer is prohibited.2. In Part 1 (Problems 1-12), mark the correct choice on your ScanTron using a No. 2 pencil. Foryour own records, also record your choices on your exam!3. In Part 2 (Problems 13-17), present your solutions in the space provided. Show all your workneatly and concisely and clearly indicate your final answer. You will be graded not merely onthe final answer, but also on the quality and correctness of the work leading up to it.4. Be sure to write your name, section and version letter of the exam on the ScanTron form.THE AGGIE CODE OF HONOR"An Aggie does not lie, cheat, or steal, or tolerate those who do."Signature:DO NOT WRITE BELOW!QuestionPoints Awarded Points1-12 4813 1214 1515 816 917 8TOTAL 10011. Find the limit: limx→0ex− cos x − 2xx2− 2x.(a) 1(b) −12(c) 0(d)12(e) Limit does no t e xist2. Solve the equation ln x + ln(x + 1) = ln(x + 4) for x.(a) x = 0 and x = 3(b) x = 4 only(c) x = 2 and x = −2(d) x = 3 only(e) x = 2 only3. Which graph of f be low has the prope rty that the derivative (f′) is always positive and decreas-ing?..(e) None of these graphs2For questions 4-5, the graph of the FIRST DERIVATIVE of a function f is shown below:4. On which interval(s) is the ORIGINAL FUNCTION f decreasing?(a) (−∞, b) ∪ (d, ∞)(b) (a, c) ∪(e, ∞)(c) (c, ∞)(d) (b, d)(e) None of these5. At what value(s) of x does the ORIGINAL FUNCTION f have a local minimum?(a) x = a, x = c, and x = e(b) x = c only(c) x = a a nd x = e(d) x = b only(e) x = b and x = d6. Find the value of log418.(a) −132(b) −23(c) −12(d) −2(e) −3237. A bacteria culture starts with 200 bacteria and triples in size every half hour. Assuming expo-nential growth, how many bacteria are there after 4 5 minutes (ignore any appropriate ro unding)?(a) 600√3(b) 750(c) 400√2(d) 1200 ln32(e) 800 ln 38. Find the value of cos(tan−14).(a)1√17(b)4√17(c)1√15(d)4√15(e)√1549. Find the absolute maximum a nd absolute minimum values of the function f (x) = x3− 3x + 1on the interval [−1, 3].(a) minimum value = −4, maximum value = 20(b) minimum value = −1, maximum value = 3(c) minimum value = −1, maximum value = 19(d) minimum value = 3, maximum value = 19(e) minimum value = −8, maximum value = 10410. Which of the fo llowing is an antiderivative of f(x) = ln x?(a)12(ln x)2(b) ex(c)1x(d) x ln x + x(e) x ln x − x11. If f (x) = 5x, what is limh→0f(x + h) − f (x)h?(a) x5x−1(b) (ln 5)5x(c) Does no t exist(d) 5x(e)5xx12. The acceleration of a car is given by a(t) = 3t + 2 (in ft/sec2). If the car is at rest at time t = 0,what is the car’s velo c ity when t = 2?(a)134ft/sec(b) 8 ft/sec(c) 12 ft/sec(d)92ft/sec(e) 10 ft/sec5PART II WORK OUTDirections: Present your solutions in the space provided. Show all your work neatly andconcisely and Box your final answer. You will be graded not merely on the final answer, but alsoon the quality a nd c orrectness of the work leading up to it.13. (6 points each)(a) Find and simplify the der ivative of f(x) = x arctan x −12ln(1 + x2).(b) If g(x) = ln |3x + 2 + e4x|, find g′(0).614. The derivative of a function f is given by f′(x) = (x − 2)e3x.(a) (4 points) Find the intervals where the original function f is increasing or decrea sing.(b) (3 points) List and classify (as max or min) the x-coordinates of all local extrema of theoriginal function f .(c) (8 points) Find the intervals where the original function f is concave upward or concavedownward.715. (8 points) A rectangular container with no top and a square bottom is to have a volume of8 ft3. Material for the sides co sts $1 per ft2and mater ial for the bottom costs $4 per ft2. Findthe dimensions that will minimize the cost of the container. Clearly show that your answer isindeed a minimum.816. (9 points) The region that lies under the graph of f(x) = cos2x from x = 0 to x =π3is shownbelow:(a) Using sigma notation, write an expression to approximate the are a under the graph ofy = f(x) with rectangles using a partition P =n0,π4,π3owith x∗ibeing the right endpointof each subinterval.(b) Evaluate the rectangle area ex pression in part (a). Your answer does not have to be sim-plified, but all trig expres sions which can be evaluated must be.(c) On the graph above, sketch the approximating rectangles.917. (8 points) Find the limit: limx→0(cos
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