Math 151 WIR, Spring 2014, c Benjamin Aurispa
Sections 4.5, 4.6, & 4.8
1. A bacteria culture starts with 2000 bacteria and grows at a rate proportional to its size. The population has grown to 2400 after 20 minutes.
(a) Find a function that models the number of bacteria after t minutes, assuming the population grows at a rate proportional to the number of bacteria.
(b) How many bacteria are present after 2 hours?
(c) When will there be 20,000 bacteria?
(d) At what rate is the population growing after 40 minutes?
2. The half-life of a radioactive substance is 10 days. How long will it take a sample of this substance to be 1/3 its original size?
3. If a sample of a radioactive substance decays to 60% of its original amount after 4 hours, what is the half-life of the substance?
4. An object with temperature 150
◦
F is placed into a room with temperature 80 the temperature of the object is 120
◦
◦
. After 20 minutes,
F. Find a function that models the temperature of the object after t minutes.
5. Suppose the rate of growth of a bacteria culture is always 5 times the current population. If there are
4000 bacteria after 2 minutes, find a function that models the population after t minutes.
6. Evaluate the following.
(a) arcsin −
√
2
2
(b) sin
(c) sin
− 1
− 1
(sin
(sin
(d) sin(arcsin
(e) arcsin(sin
π
3
)
5 π
6
)
1
4
)
9 π
8
)
(f) arccos − 1
2
(g) cos(arccos
(h) cos
− 1 (cos
5
4
)
5 π
4
)
(i) arccos(cos
20 π
11
)
(j) tan
− 1 1
3
(k) tan(tan
− 1
18)
(l) arctan(tan
2 π
3
)
(m) arctan(tan
(n) cos(arcsin
17 π
7
)
5
6
)
(o) sin(2 arctan( − 5))
(p) tan(cos
− 1 x )
1

Math 151 WIR, Spring 2014, c Benjamin Aurispa
7. Calculate the following limits:
(a) lim x →∞ sin
− 1 x
2
+ 3
2 x 2 − 5
!
(b) lim x →∞ tan
− 1 x
2
4 − x
!
8. What is the domain of f ( x ) = arcsin(4 x − 1)?
9. Calculate the derivatives for the following functions.
(a) f ( x ) = x arcsin(
√ x )
(b) g ( x ) = tan
− 1
(3 x
2
)
5
10. Find the equation of the tangent line to y = cos
− 1
(
1 x
) at the point where x = 2.
11. Calculate the following limits.
(c) lim x → 1 x
2
+ 3 x − 4
(a) lim x → 1
(b) lim x → 0
4 2 x + ln x − 16 sin x − x
1 x 3
− ln x x
1
− 1
(d) lim x →∞
( xe
1 /x
− x )
(e) lim x →∞ e
− x
(ln x )
2
(f) lim x → 0 + cot x ln(1 + 3 x + 5 x
2
) x
3
(g) lim x →∞
2
1 + x 3
3
+ x 4
(h) lim x →∞
(4 + e
3 x
)
− 2 /x
2