Chapter 4 Test Review
Page 256: 3, 17, 23, 27, 35, 37, 39, 49, 65
Find the intervals on which the function is (a) increasing, (b) decreasing,
(c) concave up, (d) concave down. Then find any (e) local extreme values,
and (f) inflection points.
3. y  x e
2
1
x2
2 x 2
yx e
dy
2
x 2
3
x 2
 x  e   2 x   e  2 x 
dx
dy
x 2
 e x 2  2 x 3   2 x 
dx
dy
x 2
 e  2 x 1  2 x 
dx
Critical values: Where does the
dy
derivative equal zero, or where is
1 x 2
2
 2 x e 1  x 
the derivative undefined?
dx
x  0 or x  1 or x  1


Find the intervals on which the function is (a) increasing, (b) decreasing,
(c) concave up, (d) concave down. Then find any (e) local extreme values,
and (f) inflection points.
3. y  x 2 e

dy
1 x 2
 2 x e 1  x 2
dx
1
x2
x  0 or x  1 or x  1
y
y'
Dec
0
Inc
1
0
Dec
0
0
Inc
1
0
a  1, 0  1, 
b  , 1  0, 1
e Local(andabsolute)minimaat 1, e and 1, e

Find the intervals on which the function is (a) increasing, (b) decreasing,
(c) concave up, (d) concave down. Then find any (e) local extreme values,
and (f) inflection points.
1
x2
3. y  x e
dy
1 x 2
c  , 0  0, 
 2 x e 1  x 2
dx
d No where
dy
x 2
1
 e 2x  2x
dx
f  None
d2y
x 2
2
1
x 2
3

e

2

2
x

2
x

2
x

e


2
x
dx2
d2y
x 2
2
2
4
y

e
2

2
x

4
x

4
x
Up
Up
2
dx
d2y
x 2
2
4
0 0

2
e
1

x

2
x

0
y' '
dx2
2





 





Critical at x  0


Use the derivative of the function y = f (x) to find the point at which f has a
(a) local maximum, (b) local minimum, or (c) point of inflection.
17. y'  6x 1x  2
Critical values: Where does the
derivative equal zero, or where is
the derivative undefined?
2
x  1 or x  2
y
Dec
y'
0
Inc
1
0
Inc
2
0
(a) No local maximum
(b) Local maximum (and absolute) minimum at x = –1.
Use the derivative of the function y = f (x) to find the point at which f has a
(a) local maximum, (b) local minimum, or (c) point of inflection.
17. y'  6x 1x  2
2
y' '  6x 1 2x  21  x  2  6
y' '  18xx  2
y Up
Up
Down
2
y' '
0
0
0
2
0
(c) Points of Inflection at x = 0 and x = 2
Find the function with the given derivative whose graph passes through the
point P.
23.
f ' x   sin x  cos x, P , 3
f x    cos x  sin x  C
f     cos  sin   C
3   cos   sin   C
3  1  0  C
2C
f x    cos x  sin x  2
Find the linearization L(x) of f (x) at x = a.
27.
f  x   tan x, a  
f ' x   sec2 x

4
 
2  
f '     sec     2
 4
 4
 
 
f     tan    1
 4
 4
   
Lx   2 x        1
  4 


Lx   2 x    1
4


L x   2 x   1
2
35.
Connecting f and f ’
The graph of f ‘ is shown in Exercise 33. Sketch a possible
graph of f given that it is continuous with domain [–3, 2] and
f(–3) = 0.
37.
Mean Value Theorem
Let f (x) = x lnx.
(a) The function is continuous on the interval [0.5, 3] and
differentiable over (0.5, 3).
(b)
1
f ' x   x   ln x 1
x
f ' x  1  ln x
f b   f a 
f ' c  
ba
f 3  f 0.5
f ' c  
3  0. 5
3 ln 3  0.5ln0.5
1  ln c 
2.5
37.
Mean Value Theorem
3 ln 3  0.5ln0.5
1  ln c 
2.5
   
   
ln c
27
27


ln
2  ln 54  2 


2 ln1 2  
2 22  2   1 ln c


ln c 
 1 1
55 5
2
c
ln 3  ln0.5

2.5
0.5
3

1

 0.4  ln 27 2  1
0.4ln27 2 1
e
c  1.579
37.
Mean Value Theorem
(c)
Secant line AB
A  a, f a   0.5, ln0.5
B  b, f b   3, ln 3
ln 3  ln0.5
 1.457
m
3  0. 5
y  ln 3  1.457x  3
y  1.457x  1.075
37.
Mean Value Theorem
(d)
Tangent line parallel to AB
c, f c  1.579, ln1.579
y  ln1.579  1.457x  1.579
y  1.457x  1.579
39.
Approximating Functions
Let f be a function with f’ (x) = sin x2 and f (0) = –1.
(a) Find the linearization of f at x = 0.
f ' 0  sin 02  0
Lx   0x  0   1
Lx  1
(b) Approximate the value of f at x = 0.1.
f 0.1  L0.1  1
39.
Approximating Functions
Let f be a function with f’ (x) = sin x2 and f (0) = –1.
(c) Is the actual value of f at x = 0.1 greater than or less than the
approximation in (b)?
Greater than the approximation since f ’(x) is actually positive
over the interval (0, 0.1) and the estimate is based on the
derivative being 0.
49.
Inscribing a Cylinder
Find the height and radius of the largest right circular cylinder
that can be put into a sphere of radius 3 as shown.
2
r
V  r h

h2 
V    3   h
4

h3
V  3h 
4
dV
3h 2
 3 
dh
4
2
2
 
2
h
   3
2
2
h
r2  3 
4
3h
3 
0
4
3h 2
3 
4
2
2
dV
3h
4

h

2
dh
2
2h
2
h
3
r
2
h
2
r  3
4
22
r  3
4
r 2
65.
Estimating Change
Write a formula that estimates the change that occurs in the
volume of a right circular cone when the radius changes from
a to a + dr and the height does not change.
dV 2
  rh
dr 3
2
dV   rh  dr
3
h
r
1
V   r 2h
3
When the radius changes from a to a + dr, the volume
change is approximately
2
dV   ah  dr .
3