Week in Review, Math 151, 4.3-4.6, 4.8, 5.1
1.Solve each equation for all possible values of x.
a)
log 2 ( 6 x  4 )  3
b ) log 2 ( x  4 )  log 2 ( x  1)  4
c) 4 2 x  e 5
2. Find the derivative with respect to x of each function.
f ( x )  tan
a)
1
x
b)
x
f ( x )  arctan(
)
and simplify
2
x
f ( x )  arcsin(
c)
)
and simplify
f ( x)  x
d)
x
2
e)
f ( x)  x
ln x
3
3. The weight of a bacteria culture doubles every 30 minutes. If initially the culture weighs 28 grams,
how long must it grow before it weighs 100 grams?
4. Polonium 210 has a half life of 140 days. If initially 500 mg are present, how many mg will be present
after t days? At how many days will it weigh 25 mg?
5. Find the derivative with respect to x of each function.
( 3 x  1)
f ( x) 
a)
x5
2 3
2 x  3 (4 x  2)
2
b) f ( x )  ln(sec
5
2
x)
6. Find the equation of the tangent line to f(x) at (a, f(a)).
3
i) f ( x )  ln( x 2 e x )
a 1
ii) f ( x )  arc sec
x
iii) f ( x )  arctan x
a  2
a 
1
3
7. Simplify each expression.
a) sin(arccos
4
5
)
b) cos(arcsinx)
c) sec(arctanx)
d) tan(arcsinx)
8. Evaluate each limit.
a)
x 
x 0
b) lim (1  4 x ) x
3 4
x
c) lim (
x 0
x
x
d) lim
5
ln x
lim
x
e) lim arccos(
x 
x 1
1 x
2x  3
)

x
x 1

1
ln x
x
f) lim x
x 0
)
9. On what intervals is the function increasing? decreasing? concave up? concave down?
a) f ( x )  ln( x
2
 1)
b) f ( x )  e
x
2
10. given the graph of f’(x) be able to tell where f is increasing/decreasing.