151 WebCalc Fall 2002-copyright Joe Kahlig
October 31, 2002
MATH 151 Section Name:
1
Show all your work.
1. Write as a single logarithm.
4 ln( x
+ 5)
−
3 ln y
+ ln z ln( x
+ 5)
4 − ln y 3
+ ln z ln
( x
+ 5)
4 y 3
+ ln ln z
( x
+ 5)
4 y 3 z
2. The population of crickets, after being sprayed with an insecticide, is given by y
= 5000 e − 0 .
25 x
, where x is measured in days since the spray was applied. How long until there are half of the crickets left? Hint: the initial population was when time is zero, i.e. x=0.
2500 = 5000 e − 0 .
25 x
0
.
5 = e − .
25 x ln(0
.
5) =
−
0
.
25 x x
= ln(0
.
5)
−
0
.
25
= 2
.
7725887 days
3. Take the derivative of the following. you do not need to simplify.
y
= ln( x 3
( x 4
+ 6 x
+ 7)
2
) y
= ln( x 3
) + ln( x 4 y
= 3 ln( x
) + 2 ln( x
+ 6 x
+ 7)
2
4
+ 6 x
+ 7) y 0
= 3
1 x
+ 2
4 x 3
+ 6 x 4
+ 6 x
+ 7
=
3 x
+
8 x 3
+ 12 x 4
+ 6 x
+ 7
4. Take the derivative and solve for y 0
. do not simplify after that.
y
= x x lny
= ln x x ln y
= x ln x y y 0 y
= 1
∗ ln x
+ x ∗
1 x
0
= x x
1
∗ ln x
+ x ∗
1 x y 0
= x x
(ln x
+ 1)