Calculus 2, MATH 136-02, 136-03

Sample Final Exam Questions: SOLUTIONS

  1. (a) A(0) = 0, A(3) = 5, A(5) = 5.     (b) A'(1) = 2.     (c) Does Not Exist (corner).     (d) Increasing: 0 < x < 4, Decreasing: 4 < x < 5.     (e) Concave Up: 0 < x < 1, Concave Down: 3 < x < 5. (f) Try plotting the points found in part (a) as well as A(1) = 1, A(2) = 3 and A(4) = 6. Then connect the dots while keeping in mind where A is increasing, decreasing, concave up or down.

  2. (a) (1/3)(2x + 1)3/2 - (1/3)cos(3x) + c     (b) (1/2)etan(2θ) + c     (c) (t7/7)(ln t - 1/7) + c     (d) (1/2)(sin-1z - z sqrt{1-z2}) + c     (e) -6ln|x| + 3ln|x-1| + 3ln|x+1| + c

  3. (a) -0.179102.     (b) -0.233891.     (c) -0.197365.   Be sure to set your calculator to radians!

  4. (a) The region lies above the parabola and below the square root curve, between x = 0 and x = 1. Its area is 1/3.     (b) 3π/10. Use the washer method.     (c) 49π/30. Use cylindrical shells.

  5. (a) an = (-1)n/2n-1.     (b) converges to ln(e) = 1.     (c) 27/2 or 13.5.     (d) [-4,4] or -4 ≤ x ≤ 4.

  6. (a) diverges by the Comparison Test (compare with the Harmonic Series).   (b) converges by a combination of the Absolute Convergence Test, Comparison Test, and p-series test.   (c) converges by the Ratio Test (r = 0).   (d) converges by the Alternating Series Test.   (e) converges by the Integral Test (the integral converges to 3/4 e-2.)

  7. (a) y = 3etan-1t.     (b) y = -ln(2/3 x3 - 9).

  8. (a) 4.4688925     (b) y = -1/(t2 + 2t - 2)     (c) y(1) = -1, so the error is approximately 5.469, which is a large error. The reason that Euler's method is so far off is that the actual solution has a vertical asymptote around t = 0.732, which happens before we reach t = 1. We say that the solution "blows up in finite time."

  9. The turkey is ready at 5:31 pm so Auntie Pat will need to serve some hors d'ouvres for half an hour (k ≈ -0.044806).

  10. (a) -2π (FTC part 2 and the chain rule).     (b) Solve the equation for y, and then rotate y = sqrt{r^2 - x^2} about the x-axis from x = -r to x = r. The volume is V = (4/3) π r3.     (c) 1.     (d) C = 3/4.     (e) FALSE. The Harmonic Series is a great counterexample.     (f) 1/2 (Final Jeopardy!) First find a formula for the terms in the series, e.g., an = 1/(n(n+1)), starting with n = 2. Then use partial fractions to write each term as a difference and cancel out the common terms.