Math 151 Week in Review
Monday Nov. 8, 2010
Instructor: Jenn Whitfield
Thanks to Amy Austin for contributing some problems.
All prolbems in this set are copywrited
Section 4.5
• Growth (k > 1) and Decay (0 < k <
1)
If the rate of change of y is proportional
dy
= ky.
to its size then
dt
Solution: y(t) = y(0)ekt
• Newton’s Law of Cooling
The rate of cooling of an object is proportional to the temperature difference
between the object and its surroundings,
provided that this difference is not too
large.
dy
= k(y(t) − Ts ), where Ts = the temdt
perature of the surroundings.
Solution: y(t) = [y(0) − Ts ]ekt + Ts
• Mixture Problems
If y(t) is the amount of substance in solution then the rate of change of the subdy
=rate in - rate out.
stance is
dt
1. A population of bacteria triples every 5 minutes. If the population follows the exponential growth model, y = y(0)ekt , find k and
then find y(t).
2. Polonium-218 is a radioactive substance
which decays exponentially. Suppose you
have 1000 mg initially, and after 5 days you
have 300 mg. How much Polonium-218 will
you have after 5 more days (10 days total)?
3. Amyium-210 has a half-life of 140 days. If a
sample has a mass of 200 mg, find the mass
after 100 days.
4. Newtons Law of Cooling states the rate of
cooling of an object is proportional to the
temperature difference between the object
and the temperature of the objects surroundings. A pie is taken from an oven
where its temperature has reached 375◦ F
and is placed on a table in a room where
the temperature is 75◦ F. If the temperature of the pie is 200◦ F after 20 minutes,
find a formula for the temperature of the pie
at time t, where t is measured in minutes.
5. A thermometer is taken from a room where
the temperature is 20◦ C to the outdoors
where the temperature is 5◦ C. After one
minute, the temperature reads 12◦ C. What
will the temperature of the object be after
2 minutes?
6. A tank contains 1500 liters of brine with a
concentration of 0.3 kg of salt per liter.
(a) Pure water enters the tank at a rate of
20 liters per minute. The solution is
kept mixed and exits the tank at the
same rate. How many kg of salt will
remain after half an hour?
(b) Brine that contains .1 kg of salt per
liter enters the tank at a rate of 20
liters per minute. The solution is kept
mixed and exits the tank at the same
rate. How many kg of salt will remain
after half an hour?
Section 4.6
• sin−1 x = y iff sin y = x and −π/2 ≤ y ≤ π/2
• cos−1 x = y iff cos y = x and 0 ≤ y ≤ π
• tan−1 x = y iff tan y = x and −π/2 ≤ y ≤ π/2
• sin−1 (sin x) = x for −π/2 ≤ x ≤ π/2
• sin(sin−1 x) = x for −1 ≤ x ≤ 1
• cos−1 (cos x) = x for 0 ≤ x ≤ π
• cos(cos−1 x) = x for −1 ≤ x ≤ 1
Derivatives of Inverse Trig Functions
d
u′
(sin−1 u) = √
dx
1 − u2
u′
d
(csc−1 u) = − √
dx
|u| u2 − 1
d
u′
(cos−1 u) = − √
dx
1 − u2
u′
d
√
(sec−1 u) =
dx
|u| u2 − 1
d
u′
(tan−1 u) =
dx
1 + u2
d
u′
(cot−1 u) = −
dx
1 + u2
Compute the following without the aid of a calculator.
1
7. arcsin
2
x→∞
x→0
2π
)
3
4
13. sin arccos(− )
5
14. cos(arctan x)
5π
)
4
16. Find the derivative of y = arctan
√
x
17. Find the equation of the tangent line to the
x
graph of y = arcsin at x = 1.
2
18. What is the domain of f (x) = arcsin(2x −
1)?
Section 4.8
INDETERMINANTE FORMS
0
∞
0 • ∞ ∞ − ∞ 00 ∞0
0
∞
1∞
L’Hospital’s Rule:
Suppose f and g are differentiable and
g′ (x) 6= 0 on an open interval I that contains a (except possible at a). Suppose
that lim f (x) = 0, lim g(x) = 0, or that
x→a
x→a
lim f (x) = ±∞, lim g(x) = ±∞. Then
x→a
x→a
f ′ (x)
f (x)
= lim ′
lim
x→a g (x)
x→a g(x)
Find the following limits using L’Hospital’s rule.
19. lim
x→1
20. lim
x→0
ln x
x−1
sin x − x
x3
(ln x)3
x→∞
x2
√
22. lim x ln x
21. lim
x→0+
3
1+
x
x
25. lim (sin x)tan x
11. tan−1 (−1)
15. arccos(cos
x→∞
24. lim
1
8. arccos √
2
√
3
9. sin−1 (−
)
2
√
10. arctan 3
12. arctan(tan
4
23. lim x x