NAME:
MATH 151
November 19, 2014
QUIZ 9
• Calculators are NOT allowed!
• Show all your work and indicate your final answer clearly. You will be graded not merely
on the final answer, but also on the work leading up to it.
1. (3 points) Evaluate the limit
lim
x→∞
3
4
1+ + 2
x x
x
.
Solution:To bring down the exponent, take natural logarithms:
x
3
4
4
3
y = 1+ + 2
⇒ x ln 1 + + 2 .
x x
x x
Taking limits and applying L’hospital’s rule:
4
3
lim ln y = lim x ln 1 + + 2
x→∞
x→∞
x x
3
ln 1 + x + x42
= lim
x→∞
1/x
(− x32 − x83 )/ 1 + x3 + x42
= lim
x→∞
−1/x2
3 + x8
=3
= lim
x→∞ 1 + 3 + 42
x
x
3
4 x
3
so y = limx→∞ 1 + x + x2 = e .
2. (3 points) Find the derivative of the following function.
−1 (πx)
f (x) = etan
Solution:Applying chain rule and the fact that
d
dx
tan−1 (πx) =
d
tan−1 (πx)
dx
1
d
−1
= etan (πx) ·
· (πx)
2
1 + (πx) dx
π
−1
= etan (πx)
.
1 + (πx)2
−1 (πx)
f 0 (x) = etan
·
1
1+x2
gives
NAME:
3. (3 points) Make a hand turkey.
MATH 151
November 19, 2014