MATH 151, FALL 2008COMMON EXAM II - 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 1-10), mark the correct choice on your ScanTron using a No. 2 pencil. For your own records,also record your choices on your exam!3. In Part 2 (Problems 11-16), present your solutions in the space provided. Show all your work neatly and conciselyand clearly indicate your final answer. Yo u will be graded not merely on the final answer, but also on the qualityand correctness of the work leading up to it.4. Be sure to write your name, section n umber and version letter of the exam on the ScanTron form.THE AGGIE CODE OF HO NOR“An Aggie does not lie, cheat or steal, or tolerate those who do.”Signature:DO NOT WRITE BELOW!QuestionPoints Awarded Points1-10 4011 1212 1013 1014 1215 816 81001PART I: Multiple Choice1. (4 pts) In order to solve the equation x5− 2x + 5 = 0, we apply Newton’s Method with an initial guess x1= 1.What value does Newton’s Method give for x2, the second approximation?(a)73(b)14(c) −13(d)74(e) −142. (4 pts) limθ→0sin2(3θ)θ2=(a) 9(b) 3(c)19(d)13(e) The limit does not exist23. (4 pts) Find the tange nt vector of unit length for r(t) =e2t, t cos tat t = 0.(a)1√2,1√2(b) h2, 1i(c) h1, 0i(d)2√5,1√5(e) h1, 1i4. (4 pts) Solve the equation ln(x + e) + ln(x − e) = 2 + ln 3.(a) x = 3e only(b) x = 1 and x = 3e(c) x = 2e only(d) x = 2e and x = −2e(e) No solution35. (4 pts) If g is the inverse of f , find g′(2) if it is known that f (3) = 2, f′(3) = 7, f (2) = 11 and f′(11) = 8. Assumeg to be differentiable.(a)17(b)111(c)18(d)12(e)156. (4 pts) If h(x) = f ◦ g = f (g(x)), find h′(−3) given that g′(−3) = 4, f′(−3) = 7, g(−3) = −2, f′(−2) = 11, andf′(4) = −3(a) 28(b) 44(c) −14(d) −6(e) −347. (4 pts) An object is moving with position function f(t) = 2 sin t−3 cos t. Find the velocity, v(t), and the acceleration,a(t), at t =π6.(a) vπ6= −√3 −32aπ6= −1 +32(b) vπ6=√3 −32aπ6= 1 −3√32(c) vπ6=√3 +32aπ6= −1 +3√32(d) vπ6=√3 −√32aπ6= 1 −3√32(e) vπ6= 1 +3√32aπ6= −√3 +328. (4 pts) If Q(x) is the quadratic appr oximation for f (x) =2xat x = 1, then Q12=(a) 3(b)52(c)32(d)72(e)9259. (4 pts) Evaluate limx→0−e1/x(a) 1(b) 0(c) ∞(d) −∞(e) e10. (4 pts) Find the inverse function of f(x) =1 − x4x + 3(a) f−1(x) =1 − 3x4x + 1(b) f−1(x) =3x − 14x + 1(c) f−1(x) =4x + 31 − x(d) f−1(x) =1 − 3x4x(e) f−1(x) =3x − 14x6PART II WORK OUTDirections: Pres e nt your solutions in the space provided. Show all your work neatly and concisely and Box yourfinal answer. You will be gra ded not merely on the final answer, but also on the quality and co rrectness of the workleading up to it.11. Find the derivative of:(i) (6 pts) f (x) = tan3(x) + tan(x3)(ii) (6 pts) g(t) =p1 +√t.712. (10 pts) Water is poured into a conical cup at the rate o f32cubic inches per second. If the cup is 6 inches tall andthe to p of the cup has a radius o f 2 inches, how fast doe s the water level rise when the water is 4 inches deep? Besure to include units with your answer. NOTE: The volume of a cone is V =13πr2h.813. (10 pts) Find the equation of the tangent line to the curve y2sin 2x = 8 − 2 y at the pointπ4, 2.914. Consider the curve given by parametric equations x = t2− 10t, y = t3− 3t2.(i) (6 pts) Find the e quation of the tangent line at t = 1.(ii) (6 pts) Find all points on the curve where the tangent line is:(a) vertical(b) horizontalExam continues on next page1015. (8 pts) Use differentials or a linear approximation to approximate√16.03.16. (8 pts) Find all value(s) of x, 0 ≤ x ≤ 2π, where f (x) = x + 2 sin x has a hor izontal tangent.End of
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