MATH 104 – Practice Final Exam Answers
1. y= 100,000,000 ·10−2x
2. 2 −π
4
3. 2π
5(9 −ln 10)
4. y=x−1/(4x4)e1
16 (1−
1
x4)−1
5. The differential equation is dy
dx =k(M−y), with initial condition y(0) = 0. If y(2) = 10, then
M/2 of the words are learned at time t=2 ln 2
ln M−ln(M−10) .
6. e−2
7. π
4+ln 2
2
8. 1
3ln(ex−4) −1
3ln(ex−1) + C
9. 2
7−√2
56
10. Vertical asymptote (toward −∞) at x= 0, x-intercept at x= 1, Max at (e, 1/e), Inflection at
(e3/2,3/(2e3/2)), horizontal asymptote y= 0.
11. 4√15 −ln(4 −√15)
12. C= 1, integral is ln8/3 (don’t forget x= 0!)
13. 1
14. Converges absolutely (by the root test)
15. The series is alternating and starts out 0.1−(0.1)5
10 +···. So we can just use 1 term, and the
answer is 0.1. (To a bunch of decimal places the integral is 0.099999000041664)
16. Series converges in the interval [19/4, 21/4). Only conditionally at the left endpoint.
17. 2 + 1
32 (x−16) −3
4096 (x−16)2+7
262144 (x−16)3. If you put x= 15 into this, the series does not
alternate, so you must use Lagrange’s form of the remainder. The error is about 6.5×10−6.
18. x2
2! −x3
3! +x4
4! −x5
5!
19. Draw a picture, then work the integrals and exponentiate both sides.
