MATH 205 Final Exam Solutions Name: 1. (12 points) The keratoid ...

MATH 205 Final Exam Solutions Name:
7. (12 points) Evaluate the following integrals:
(a) (4 points) ∫ 2
0
sec √ x tan √ x
√ x
dx.
Since we have sec √ x tan √ x in the integrand, it makes sense to use the substitution
u = √ x to simplify this expression to sec u tan u. We are particularly fortunate since then
du
= 1
dx 2 √ and thus 2du = √1
x x
dx, and a division by √ x is in fact present in our integral.
We may now perform the u-substitution:
∫ 2
0
sec √ x tan √ x
√ x
dx =
(b) (4 points) ∫ 1
−2 2x2 + 3
x 2 +1 dx.
Using known antiderivatives,
∫ 1
−2
∫ √ 2
0
√
sec u tan u(2du) = 2 sec u] 2
0
= 2 sec √ 2 − 2 sec 0
2x 2 + 3 [ 1 2
x 2 + 1 dx = 3 x3 + 3 arctan x]
= 2 ( )
−16
−2
3 + 3 arctan 1 − 3 + 3 arctan(−2)
(c) (4 points) ∫ 1
x ln x dx.
1
Since we have in the integrand, it makes sense to use the substitution u = ln x to
ln x
simplify this expression to 1 du
. We are particularly fortunate since then = 1 and thus
u dx x
du = 1 dx, and a division by x is in fact present in our integral. We may now perform the
x
u-substitution: ∫
∫
1 1
x ln x dx = du = ln u + C = ln(ln x) + C
u
8. (12 points) Determine the following limits.
(a) (4 points) Evaluate lim x→0 (e x − 1) csc x or demonstrate that it cannot be evaluated.
This is a 0 × ∞ form which can be rephrased as the 0 indeterminate form lim 0 x→0 ex −1
. sin x
Applying L’Hôpital’s rule:
e x − 1
lim
x→0 sin x = lim e x
x→0 cos x = 1 1 = 1
(b) (4 points) Using the difference quotient, find the derivative with respect to x of f(x) =
4x 2 − x + 5. You may not use L’Hôpital’s rule for this problem.
f ′ (x) = lim
h→0
f(x + h) − f(x)
h
(4(x + h) 2 − (x + h) + 5) − (4x 2 − x + 5)
= lim
h→0 h
(4x 2 + 8xh + 4h 2 − x − h + 5) − (4x 2 − x + 5)
= lim
h→0 h
8xh + 4h 2 − h
= lim
h→0 h
= lim 8x + 4h − 1 = 8x − 1
h→0
Page 4 of 6 November 30, 2011