Week in Review # 5
b.
MATH 142
Section 4.4, 4.5, & 5.1
f ( x)
c.
f  x
f  x  0
f ( x)  c
b. f  x   x
2
Drost-Fall 2014
Oct 13, 2014
Derivative Rules
a.
g  x   5x  ln 10  2 x
f   x   n  x n 1
n
f ( x)  (mess ) n
f ( x)  n  mess 
n 1
 mess 
f ( x)  g ( x)  h( x)
f ( x)  g ( x)  h( x)  h( x)  g ( x)
g ( x)
e. f ( x) 
h( x )
h( x)  g ( x)  g ( x)  f ( x)
f ( x) 
[h( x)]2
x
f. f ( x )  e
f   x   ex
d.
g. f ( x)  e
c.
h  x 
d.
F  x   e x  ln x
e.
 3x  4 
G ( x)  ln
f ( x)  a x  ln a
f ( x)  a x
i.
f ( x)  a mess
f   x   a mess  mess  ln a
j.
f ( x)  ln x
1
x
mess
f ( x) 
mess
1
f ( x) 
x ln b
mess
f ( x) 
mess  ln b
k. f ( x)  ln  mess 
l. f ( x)  log b x
m. f ( x)  log b  mess 
f ( x) 
Use logarithmic rules to rewrite BEFORE finding the
derivative.
M
 ln M  ln N  ln P
NP
1. Find the derivative of each of the following
functions.
a.
e x  e x
f ( x)  mess  e mess
mess
h.
ln
ln  x 2 
f  x   4 3  log 2 x3  5 x  e6 x
2
5
23 x
 e2 x 5
2. Find the x-value(s) where
the derivative is zero:
a.
y  x 4  e2 x
b.
y   ln x 
 kt
y  ae be , where
a, b, k  0 Show the derivative
b.
is always positive.
4
5.
Given the demand function
20p + x = 800,
a. Find the elasticity of demand, E(p).
3. Suppose the profit function
is given by
P  x   x  e0.2 x ;
2
find the marginal profit and
where the marginal profit is
zero.
b. Find E(15) and determine if the price
should be raised, lowered, or left the same.
c. If the $15 price changes by 10%, what is
the approximate change in demand?
4. Find the derivatives of:

a. y  ln ln x  e
4
2

d. What price maximizes revenue?
6. Where is the function,
decreasing?
, increasing or
7. The concentration of pain killer in the bloodstream
t hours after taking the medicine is given by
C (t ) 
t  9
2
t  4t  10
2
, where C(t) is measured in
9.
Find the equation of the tangent to the curve:

f ( x)  4 x  8

3
at
x9.
10. Find the derivative:
a. f  x   8
6x
mg/ml. How many minutes before the pain− killer has
reached its maximum concentration?
b. g ( x )  log 5 (12 x)
c.
y  2e3 x  6
d.
y  ln[( f ( x))3   g ( x)  ]
8. Find the intervals over which f(x) is increasing
when
f ( x) 
x6
2
x  x6
2
11. Plasma Plus determines that the price-demand
function for their newest 27” screen is
p( x)  
12. Given the graph of f(x):
x
 500 , where x represents the
400
6
4
number of screens produced and sold. They have fixed
costs of $1797.75 and it cost the company $495 to
make each screen.
2
−4
−2
2
4
6
a. Find the revenue function, R(x), and the cost
function, C(x).
a. Find the critical values.
b. State the intervals over which the function is
increasing, decreasing, and constant.
c. Give the points of any extrema.
b. Find the profit function, P(x) and find the smallest
and largest production levels x so that the company
realizes a profit.
13. Given the graph of f   x  :
a. State the critical values
6
of
f ( x).
4
2
−4
c. Evaluate P  500  and interpret.
−2
2
4
6
b. State the intervals
over which f ( x) is
increasing, or constant.
c. State the x-values of any
relative extrema on f ( x).
d. How many screens should they make and sell to
maximize profits?
14. Find the equation of the tangent to the curve
f(x) = x2 lnx at x = e.
·
15. Find the derivative of f(x) = 5x ln(x2 + 10).
16. Find the derivative of
g ( x) 
19. Using calculus, determine the critical values, the
intervals where the function is increasing or decreasing,
and any points of relative extrema.
e x
log 2  x 2  5 x 
a.
f ( x) 
1
x 9
b.
g ( x) 
ln x 2
x
2
DO NOT SIMPLIFY.
17. Find the derivative and DO NOT SIMPLIFY:
c. h( x)  e
20.
2 x
18. Given f ( x )  12 x  e
,
a. find the relative rate of change.
b. find the percentage rate of change.
Given
2x
 e 2 x
f ( x) is a continuous function, with
f ( x)  n  x( x  4)( x  2) 2 ( x  3)3 where n is a
negative function, determine the critical values of f ( x) ,
the intervals where f ( x) is increasing or decreasing
and any values of x where relative extrema will occur.