(b) Show that does not exist.
(c) Show that has a vertical tangent line at .
(Recall the shape of the graph of . See Figure 13 in Sec-
tion 1.2.)
48. (a) If , show that does not exist.
(b) If , find .
(c) Show that has a vertical tangent line at .
;
(d) Illustrate part (c) by graphing .
Show that the function is not differentiable
at 6. Find a formula for and sketch its graph.
50. Where is the greatest integer function not differ-
entiable? Find a formula for and sketch its graph.
(a) Sketch the graph of the function .
(b) For what values of is differentiable?
(c) Find a formula for .
52. The left-hand and right-hand derivatives of at are
defined by
and
if these limits exist. Then exists if and only if these one-
sided derivatives exist and are equal.
(a) Find and for the function
(b) Sketch the graph of .
(c) Where is discontinuous?
(d) Where is not differentiable?
53. Recall that a function is called even if for all
in its domain and odd if for all such .
Prove each of the following.
(a) The derivative of an even function is an odd function.
(b) The derivative of an odd function is an even function.
54. When you turn on a hot-water faucet, the temperature of
the water depends on how long the water has been running.
(a) Sketch a possible graph of as a function of the time
that has elapsed since the faucet was turned on.
(b) Describe how the rate of change of with respect to
varies as increases.
(c) Sketch a graph of the derivative of .
55. Let be the tangent line to the parabola at the point
. The angle of inclination of is the angle that
makes with the positive direction of the -axis. Calculate
correct to the nearest degree.
&
x
!
&
!!1, 1"
y ! x
2
!
T
t
tT
tT
T
xf !"x" ! "f !x"x
f !"x" ! f !x"f
f
f
f
1
5 " x
if x ' 4
f !x" !
0
5 " x
if x ( 0
if 0
)
x
)
4
f #
!
!4"f #
"
!4"
f #!a"
f #
!
!a" ! lim
h
l
0
!
f !a ! h" " f !a"
h
f #
"
!a" ! lim
h
l
0
"
f !a ! h" " f !a"
h
af
f #
fx
f !x" ! x
$
x
$
51.
f #
f !x" ! % x&
f #
f !x" !
$
x " 6
$
49.
y ! x
2#3
!0, 0"y ! x
2#3
t#!a"a " 0
t#!0"t!x" ! x
2#3
f
!0, 0"y !
s
3
x
f #!0"
41. The figure shows the graphs of three functions. One is the
position function of a car, one is the velocity of the car, and
one is its acceleration. Identify each curve, and explain your
choices.
42. The figure shows the graphs of four functions. One is the
position function of a car, one is the velocity of the car, one is
its acceleration, and one is its jerk. Identify each curve, and
explain your choices.
;
43– 44 Use the definition of a derivative to find and .
Then graph , , and on a common screen and check to see if
your answers are reasonable.
43. 44.
;
If , find , , , and .
Graph , , , and on a common screen. Are the
graphs consistent with the geometric interpretations of these
derivatives?
46. (a) The graph of a position function of a car is shown, where
s is measured in feet and t in seconds. Use it to graph the
velocity and acceleration of the car. What is the accelera-
tion at t ! 10 seconds?
(b) Use the acceleration curve from part (a) to estimate the
jerk at seconds. What are the units for jerk?
47. Let .
(a) If , use Equation 3.1.5 to find .f #!a"a " 0
f !x" !
s
3
x
t ! 10