KEY
Calculus AB
Chapter 4a5Review
Chapter
Part 1 Test Review (Attn: This review DOES NOT include Section 5.6)
1. When looking for absolute extrema, where do the possible extrema exist, and how do you find them?
at
endpoints and critical points
find first derivative f
to c and see absmax ano min values
set f
o
solve for
values
plug in all
candidates
2. How do you justify relative extrema?
1stderivative test lookatvalues were fi changessign
f goesfrom
f goesfrom
ire max
2ndderivative test findwure t
is
o orone andfindsign off
criticalvalues
3. How do you justify that a function is increasing or decreasing?
fix
when
increases
re min
f c o rel max
f
o rel min
fi so
when f a o
far
decreases
fix
is concave up wun f
4. How do you justify that a function is concave up or concave down?
o
fix is concave down wun f Lo
5. How do you justify that a function has a point of inflection?
f
has a point of inflection when
f
o ou DNE
AND has
a
signchange at that value of
6. Using the graph of g (x) below, determine the signs of g ' x and g '' x at each point. Explain your reasoning.
At x = a …gca
At x = b …
gas
At x = c …
gus
At x = d … ca
g
decreasing
concave
horizontal tangent concaveup so glass
horizontal tangent
g (x)
up so gca co and Casso
g
o andg
so
gco o andgice co
concavedown so
decreasing concavedown
sogcascoandgcasco
7. Given the graph of f ' below answer each of the following questions, and justify your response with a statement that
contains the phrase “since f ' ____________________ …”
a) When is f increasing?
since
f
c is
b) When is f decreasing?
since f co on Cbd
o on cab V d e
increasing on
those
interval
intervals
c) When is f concave up?
d) When is f concave down?
since fi is increasing on
ce
f is concave up on
that interval
since f is decreasing on
Lac
o
o
x b and
f is
concavedown on
that interval
e) When does f have a relative maximum?
since f
f
is decreasing on that
f
f) When does f have a relative minimum?
since f
changes
t has a rel max
from
g) When does f have a point of inflection?
D
from
o
o
d and f changes
t has a rel min
a
8. Find the value of c guaranteed by the MVT for f
X
Hb
f
x
8
f
c
8ct5
5
f.cc
fu
b
4 x2
x
5 x on the interval [–2, 1].
sects i i z
sets 3
3
sects
so 4
a
442 sci 4 5 9
4C25 so23 1610 6
fc
9. [Calculator Allowed] Find the value of c guaranteed by the MVT for
X
[
f x sin x on the interval [4, 5].
: For those of you doing this problem algebraically, the answer is NOT c 1.774 …Why?]
fuss fed
f x
cost
c cosCo 2oz
f.cc
b a
cosc
zit 1.774
f c
cosc
10. Suppose y
x
3
c 1.774
sins since
s 4
cosc
f 4S_sinus
fess sines
c 4.509
butnot on interval 4,53
o2oz
3 x . [No Calculator]
a) Find the zeros of the function.
x 1 2 37 0
o
y
B
x
B
b) Determine where y is increasing or decreasing and justify your response.
2
a
i
y is increasing on C is Duct s
y 3 3 0
s
s
i
i
b c y o on use intervals
3h2 D 0
g't
He
y't
ex Dcx D o
on C i D b c
x
y
x i I
c) Determine all local extrema and justify your response.
rel max
y
i
rel min as x i
L13 3CD
yl is xD
13 2
i s
z
is decreasing
y LO on this interval
rei.max z
ci.z
zoa
rel min
2
d) Determine the points where y is concave up or concave down, and find any points of inflection.
Justify your responses.
6x
y
0 o
o
x o
t.us
y
is concave down on
y
on
bk a co
ofnc.is
is concave up
bc y o
e) Use all your information to sketch a graph of this function.
incr
i
i
v
challenges
12. If f ' x
flx
x
2
9x
I
i
r
X I
1 , what does f (x) equal?
9
t x t C
i
p.o.i.ax
co 3
dear dear
down up
down
o
Co.o
inc
up
o
fiasco fiasco f m of c
co
o
13. Suppose
d2y
dx 2
x3
4 x 2 . Justify each response below.
a) Where is y concave up?
x3 4 2 0
x o
x 43 0
b) Where is y concave down?
Co
o v
x
4
s
0 o 0 4
is
fus acid
fun
4
0,4
c) Are there any inflection points on y? If so, where?
d
pro i
x
f has
4
a
signchange
14. [Calculator Allowed] The derivative of h x is given by h ' x
Zoos x
Justify EVERY response.
age
coscx
a) Where is h (x) increasing?
5
Iu
2b
73,25
b) Where is h (x) concave down?
C5.760
2.618
V
xE
o
ta
x
E
x
at
x mo
t
x
at
0
IT
x
0.524 3.665
4 az
rel max
5.760 0.524 2.6183.665
bc
f
.
has changefrom 0
a child changefrom
3.665,2618
d) Does h (x) have a point(s) of inflection? If so, where?
changes sign
15. A rectangle is inscribed between the parabolas y 4 x and y 30 x as shown in the picture.
Hs
What is the maximum area of such a rectangle? Justify your response using CALCULUS.
2
Chap 5
part
aromas
2
2 ,2
.
x
c) Find the x-coordinates of all extrema of h (x) on the interval 2 , 2
when h x O andendpoints
zit isat
rel mina
x
1 on the interval
6
o
Eatzit
coshes o
when u
2 cos x
2
0
0
0
at
16. USE CALCULUS: Find the maximum area of a rectangle inscribed under the curve f x
Chap 5
part
aged
16
x2 .
2
17. [Calculator Allowed] USE CALCULUS: A rectangle is inscribed under one arch of y 8 cos 0.3 x with its base on
cat
the x-axis and its upper two vertices on the curve symmetric about the y-axis. What is the largest area the rectangle can
have?
arms
Chap 5
part
2
18. The function f is continuous on [0, 3] and satisfies the following:
x
f
f'
f ''
0
0
–3
0
0<x<1
Neg
Neg
Pos
1
–2
0
1
1<x<2
Neg
Pos
Pos
2
0
DNE
DNE
2<x<3
Pos
Pos
Pos
3
3
4
0
a) Find the absolute extrema of f and where they occur.
fo
f l
O
f 2
2
o
f3
3
differentiable
abs min
z
0
1 2
abs max
3 at
3,3
b) Find any points of inflection.
none no change in
sign for f
c) Sketch a possible graph of f.
3
z
i
i
Go back andzReview the questions from your assignments in this chapter … Understand your notecards!