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Math 151 WIR, Fall 2013, Benjamin
Aurispa
Math 151 Week in Review 10
Sections 4.5, 4.6, & 4.8
1. A bacteria culture starts with 2000 bacteria and quadruples every 25 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) At what time are there 30,000 bacteria?
2. The half-life of a radioactive substance is 10 days. How much of a 30 g sample remains after 2 weeks?
3. 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.
4. An object with temperature 150◦ F is placed into a room with temperature 80◦ . After 20 minutes,
the temperature of the object is 120◦ F. Find a function that models the temperature of the object
after t minutes.
5. Evaluate the following.
√ (a) arcsin −
2
2
(b) sin−1 (sin π3 )
(c) sin−1 (sin 5π
6 )
(d) sin(arcsin 14 )
(e) arccos − 12
(f) cos(arccos 45 )
(g) cos−1 (cos 5π
4 )
(h) tan−1
(i)
√1
3
−1
tan(tan 18)
(j) arctan(tan 2π
3 )
(k) cos(arcsin(− 56 ))
(l) sin(2 arctan 5)
(m) tan(cos−1 x)
6. Calculate the following limits:
−1
x2 + 3
2x2 − 5
−1
x2
4−x
(a) lim sin
x→∞
(b) lim tan
x→∞
!
!
7. What is the domain of f (x) = arcsin(4x − 1)?
8. Calculate the derivatives for the following functions.
√
(a) f (x) = x arcsin( x)
(b) g(x) = tan−1 (3x2 )
5
1
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Math 151 WIR, Fall 2013, Benjamin
Aurispa
9. Find the equation of the tangent line to y = cos−1 ( x1 ) at the point where x = 2.
10. Calculate the following limits.
x2 + 3 x − 4
x→1 42x + ln x − 16
sin x − x
(b) lim
x→0
x3
1
1
(c) lim
−
x→1 ln x
x−1
(a) lim
(d) lim (xe1/x − x)
x→∞
(e) lim e−x (ln x)2
x→∞
(f) lim cot x ln(1 + 3x + 5x2 )
x→0+
(g) lim
x→∞
1+
2
3
+ 4
3
x
x
x3
(h) lim (4 + e3x )−2/x
x→∞
2