Calculus 2 with FUNdamentals, MATH 13402
Sample Final Exam Questions: SOLUTIONS
 (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.

(a) (1/3)(2x + 1)^{3/2}  (1/3)cos(3x) + c
(b) (1/2)e^{tan(2θ)} + c
(c) (x^{7}/7)(ln x  1/7) + c
(d) sqrt{4y^{2}  13}/y + 2ln2y + sqrt{4y^{2}  13} + c
(e) 6lnx + 3lnx1 + 3lnx+1 + c

(a) (I) Righthand Sum, (II) Midpoint Sum, (III) Trapezoid Rule, (IV) Lefthand Sum.
(b) 1.253633675.

(a) 1/3.
(b) 3π/10.
(c) 3π/10.

(a) x=10, p=98.
(b) CS = $368.04, PS = $200.

(a) y = 3e^{tan1t}.
(b) y = 1 (equilibrium solution).

(a) 4.4688925
(b) y = 1/(x^{2} + 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.

The turkey is ready at 5:31 pm so Auntie Pat will need to serve some hors dâ€™ouvres for half an hour.

(a) a_{n} = (1)^{n+1}/2^{n1}.
(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 3/4e^{2}).

(a) Solve the equation for y, and then revolve y = sqrt{r^2  x^2} about the xaxis from x = r to x = r. The volume
is V = (4/3) πr^3.
(b) 52.
(c) 1.
(d) k = 2.