Spring 2007Math 151Common Exam 3Test Form APRINT: Last Name First Name:Signature: ID:Instructor’s Name: Section #INSTRUCTIONSIn Part 1 (Problems 1–12), mark the correct choice on your ScanTron form usinga #2 pencil. For your own records, also record your choices on your exam! TheScanTrons will be collected after 1 hour; they will NOT be returned.In Part 2 (Problems 13–18), write all solutions in the space provided. CLEARLYINDICATE YOUR FINAL ANSWERSNo Calculators Permitted1Math 151 Multiple Choice Part I(4 points each)1. Which graph illustrates a local maximum which is also an inflection point?(a) (b) (c)(d) (e)2. Consider the function f(x) = 3x4− 8x3+ 5. Find the local minima of f (x).(a) x = 0 and x = 2 (b) x = 0 and x =43(c) x = 1 (d) x = 2 (e) x =433. Find the derivative of f (x) = tan−1(ln x).(a) f0(x) = tan−1µ1x¶(b) f0(x) =µ1x¶1(ln x)2+ 1(c) f0(x) =1xsec−2(ln x) (d) f0(x) =xx2+ 1(e) f0(x) =1tan−1xµ1x2+ 1¶24. Consider the graph of the function f (x) = 10x6− 24x5+ 15x4. Find the x-coordinates of theinflection points.(a) x = 0 (b) x = 0 and x = 1 (c) x = 1(d) x = 1 and x =35(e) x = 0, x =35, and x = 15.50Pi=13i − 75=(a) 415 (b) −40 (c)1435(d) 23 (e) 6956. Find the region where the function f(x) = 2x5− 5x4− 10x3is increasing.(a) x > 3 (b) −1 < x < 3 (c) x < −1 or x > 3(d) x < −1 or 0 < x < 3 (e) −1 < x < 0 or x > 337. limx→01 − cos3xsin2x + sin(2x)=(a) does not exist (b) −34(c) 0 (d)32(e) 18. Find the derivative of f (x) = xln x.(a) f0(x) =1xxln x(b) f0(x) = 2ln xxxln x(c) f0(x) = x1/x(d) f0(x) =ln xxxln x(e) f0(x) = x2/x9. cosµsin−1µ−45¶¶=(a) −35(b)35(c) −34(d)34(e) does not exist410. e3 ln 2−1ln(5e2) =(a)8e(ln 5 + 2) (b)e3ln 2(ln 5)2(c) e3ln 5(d) 2µ8 +1e¶ln 5 (e)µe3ln 2 +1e¶(ln 5 + 2e)11. Which function is an anti-derivative of1√1 − x2?(a) 2√1 − x2(b) tan−1x (c) x(1 − x2)−3/2(d) sin−1x (e)1x2+ 112. Find the graph of the consistently increasing function whose derivative is consistently decreasing.(a) (b)(c) (d)(e)5Math 151 Work-Out Problems Part IIShow your work. No credit for unsupported answers will be given.13. The radioactive decay law states that the rate of decrease in the amount of a radioactive isotope isalways proportional to the amount remaining. The half-life of an isotope is the length of time overwhich the amount of the isotope is reduced by half through radioactive decay. Now suppose the initialamount of a given isotope is 7 grams and that 4 grams remain after 15 ln 7 − 30 ln 2 years.(a) Find the formula for how the amount of isotope depends on time, (4 points)(b) Find the half-life of the isotope. (No decimal approximation is required. If you obtain an answerlike 2ee2ln π, then leave it alone. 3 points)14. Calculate the following limits. (3 points each)(a) limx→0tan−1xsin x(b) limx→∞ln x3√x6(c) limx→0(1 + 2x)3/x15. Consider the function f (x) =√x and partition the interval 1 ≤ x ≤ 4 into 6 equal sub-intervals.Calculate the Riemann sum for the pointsx∗1= 1, x∗2=169, x∗3=94,x∗4=259, x∗5=4916, x∗6= 4of evaluation. (10 points)16. Fencing is required to enclose a rectangular field with 7500 square feet of area. The north side requireshigher quality fencing than the other sides. If the regular fencing costs $300per foot and the higherquality fencing costs $500per foot, how long must the north side be in order to minimize the cost?(10 points)717. A car is accelerating along a straight highway, where the acceleration itself is increasing linearly—specifically, after t seconds, the acceleration is given bya(t) = 30t + 8in feet per second per second.(a) Find the velocity v(t) as a function of t if the initial velocity is v(0) = 44 ft./sec. (5 points)(b) Find the position x(t) as a function of t if the initial position is 100 feet down the road.(5 points)18. Differentiate the function f (x) =(2x − 1)3√x + 23√5x − 1(3x + 5)2. (6 points) (Hint: The direct approach
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