Calculus 2 with FUNdamentals, MATH 134-02

Sample Final Exam Questions: SOLUTIONS

1. (a) F(0) = 0, F(3) = 5, F(5) = 5.     (b) F'(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 F(1) = 1, F(2) = 3 and F(4) = 6. Then connect the dots while keeping in mind where F 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) (x7/7)(ln x - 1/7) + c     (d) -sqrt{4y2 - 13}/y + 2ln|2y + sqrt{4y2 - 13}| + c     (e) -6ln|x| + 3ln|x-1| + 3ln|x+1| + c
3. (a) (I) Right-hand Sum, (II) Midpoint Sum, (III) Trapezoid Rule, (IV) Left-hand Sum.     (b) 1.253633675.
4. (a) 1/3.     (b) 3π/10.     (c) 3π/10.
5. (a) x=10, p=98.     (b) CS = \$368.04, PS = \$200.
6. (a) y = 3etan-1t.     (b) y = 1 (equilibrium solution).
7. (a) 4.4688925     (b) y = -1/(x2 + 2x - 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 x = 0.732, which happens before we reach x = 1.
8. The turkey is ready at 5:31 pm so Auntie Pat will need to serve some hors dâ€™ouvres for half an hour.
9. (a) an = (-1)n+1/2n-1.     (b) converges to -1/3.     (c) 27/2 or 13.5.     (d) (i) diverges by the Comparison Test (compare with the Harmonic Series).   (ii) converges since it is a Geometric series with r = 3/π < 1.   (iii) convegres by the Integral Test (the integral converges to 3/4e2).
10. (a) Solve the equation for y, and then revolve y = sqrt{r^2 - x^2} about the x-axis from x = -r to x = r. The volume is V = (4/3) πr^3.     (b) 52.     (c) 1.     (d) k = 2.