Week in Review # 7
5. Sketch the graph of a function f that satisfies the
MATH 142
Sections 5.2, 5.3, 5.4
Drost-Fall 2014
Oct 20, 2010
following:
Domain: All real numbers where
x-intercepts: (−2,0) and (2,0)
y-intercept: (0,4)
Vertical asymptotes: none
x  3
lim f ( x)  , lim 0
1. Find the intervals over which h(x) is decreasing
x 
 x 1 
AND concave up, when h( x)  ln  2
.
 x 4
2
x 
f ′(x) > 0 on (−2,0) ∪ (4,∞)
f ′(x) < 0 on (−∞,−3),(−3,−2)∪ (0,4)
f ′′(x) > 0 on (−∞,−3) ∪ (−3,−1) ∪ (1,6)
f ′′(x) < 0 on (−1,1) ∪ (6,∞)
2. Find the critical values for
6. Use the graph below to find the absolute extrema
(locations and values) of f (x) on each of the
intervals below.
a.[ − 5, 0]
a.
b.
3. From the graph below of f ′(x),
a. Where is f′′(x) > 0?
6
b.[ − 1, 2]
4
b. Where is f(x) increasing?
c. Where does f(x) have a relative max
c.[1 , 5]
or min?
d. Where does f ′(x) have a point of
d.(0 , ∞ )
2
−6 −4 −2
inflection?
e.( − ∞ , − 1)
y
f(x)
24
6
−2
−4
−6
6
4
2
−4 −2
2
4
6
x
4. Sketch the graph of a function f that satisfies the
following:
Domain: (−∞,4) ∪ (4,∞)
Vertical asymptotes: x = 4
Horizontal asymptote: y  2
+++++
−4
f ’’(x) − − − −
−−−−
0
−−−−−−− −
−4
7. f(x) = x4− 4x3− 80x2− 120
8. g(x) = −0.2x + ln(5x − 20)
9. h(x) = 2x5− 15x4− 90x3 + 75
x-intercept: (6,0); y-intercept: (0,−3)
f ’(x)
−−−−
Find all local extrema of each of the given functions
on its domain. Use the Second Derivative Test when
it applies.
−−−−−
4
+++++
4
10. a) Darcy’s Garden Center estimates that it will sell
N(x) units of a product after spending x
thousand dollars on advertising, when
N(x) = −x4 + x3 − 12x2− 2x + 10.
When is the rate of change of the product
increasing, and when is it decreasing?
b) What is the point of diminishing returns?
11. Apply the graphing strategy to sketch the
graph of
.
12. Find ALL asymptotes for each of the following
functions:
a.
b.
c.
d.
F ( x) 
3x 2  x  2
6 x3  5 x 2  6 x
13. Let S(t) = −0.005t4+0.04t3+1.8t2+25t represent
the salary you are offered on your first job after
graduation from A&M, in hundreds of dollars. Over
what intervals is the rate of change of Salary negative?
Define t as the number of years working at this job.
14. Find the limit of each of the following:
a.
b.
c.
d.
e.
3x 2  8 x  12
x 
4 x3  100 x
6 x2  4 x  5
lim
x  12 x 2  x  1
6 x3  4 x 2  5
lim
x 
x2  2 x  1
1
lim 16 
x 
x
lim
lim
x 
8  4e 
x