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Math 151 WIR, Spring 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. 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.
4. A curve has the property that at every point the slope of the curve is 5 times the y-coordinate. If the
curve passes through the point (2, 4), find the equation of the curve.
5. Evaluate the following.
√
2
2 )
arccos(− 21 )
tan−1 √13
sin−1 (sin 5π
6 )
cos(arccos 54 )
cos−1 (cos 5π
4 )
−1
tan(tan 18)
(a) arcsin(−
(b)
(c)
(d)
(e)
(f)
(g)
(h) arctan(tan 6π
7 )
(i) cos(arcsin(− 56 ))
(j) sin(2 arctan 5)
(k) tan(cos−1 x)
"
−1
6. Calculate lim sin
x→∞
x2 + 3
2x2 − 5
!
+ tan
−1
x2
4−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( 5x)
(b) g(x) = tan−1 (3x2 )
5
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
(a) lim
1
c
Math 151 WIR, Spring 2013, Benjamin
Aurispa
(c) lim
x→1
1
1
−
ln x x − 1
(d) lim (xe1/x − x)
x→∞
(e) lim e−x (ln x)2
x→∞
(f) lim cot x ln(1 + 3x + 5x2 )
x→0+
2
3
(g) lim 1 + 3 + 4
x→∞
x
x
(h) lim sin xtan x
x3 /5
x→0+
(i) lim (4 + e3x )−2/x
x→∞
2