Exercises: Slopes and Derivatives
dy
1–4 Estimate the value of
for the given curve at the given
dx
point. Your answers must be correct to two decimal places.
60,000
1. y = x3 , (2, 8)
3. y =
1
1 + x2
10. The following graph shows the population of a certain culture
of bacteria over the course of a six-hour biology experiment:
2. y = 2x(4 − x), (3, 6)
4. y = 2x , (1, 2)
, (2, 0.2)
50,000
Population
40,000
30,000
5. Consider the parabola y = 5 − x2 .
(a) Find the slope of this curve at the point (2, 1).
(b) Find the equation of the tangent line to the curve at this
point.
(c) Sketch a graph showing the parabola as well as the tangent
line you found in part (b).
6. An apple pie is removed from the oven and left to cool. The
temperature T of the pie decreases according to the formula
T = (70 ◦ F) + (0.98)t (280 ◦ F),
where t is the time in minutes.
(a) Find the average rate at which the temperature decreases
during the first 30 minutes.
(b) Estimate the value of
dT
at t = 30 minutes.
dt
20,000
10,000
0
0
Time HHoursL
1
2
3
4
5
6
(a) What was the average growth rate of the bacteria over the
course of the experiment?
(b) Estimate the growth rate of the bacteria at t = 2 hours.
(c) At what time was the bacteria population the largest?
(d) At what time was the bacteria population growing the
fastest?
11. The following graph shows the distance traveled by a cheetah
over the course of a short sprint:
300
250
7. A ball is thrown straight up into the air. The height h of the ball
above the ground in meters is given by the formula
h = 1.2 + 23t − 4.9t 2
where t is the time in seconds.
(a) What is the initial velocity of the ball?
(b) Estimate the velocity of the ball two seconds after it is
thrown.
(c) When will the ball hit the ground?
8. The radius of a sphere is increasing at a rate of 3 cm/min.
(a) Estimate the rate at which the volume of the sphere is
increasing when the radius is 10 cm.
(b) Estimate the rate at which the surface area of the sphere is
increasing at that time.
200
Distance
150
HMetersL
100
50
0
0
5
10
15
Time HSecondsL
20
(a) Find the average speed of the cheetah over the course of
the 20-second sprint.
(b) Estimate the speed of the cheetah at t = 15 seconds.
(c) Estimate the maximum speed attained by the cheetah.
12. The following graph shows the speed of a race car during the
first ten seconds of a race:
100
9. The following graph shows y as a function of x.
80
y
Speed 60
HmsL 40
1
20
-2
-1
1
2
x
-1
dy
for x = −2, x = −1,
Use this graph to estimate the values of
dx
x = 0, x = 1, and x = 2.
0
0
2
4
6
Time HsL
8
10
(a) Estimate the speed of the car at t = 4 seconds.
(b) Estimate the acceleration of the car at t = 4 seconds.
(c) Estimate the average acceleration of the car during the first
four seconds.
Answers
1. 12
2. −4
3. −0.16
4. 1.39
y
5
4
5. (a) −4 (b) y = 9 − 4x (c)
3
H2,1L
2
1
-3
-2
-1
-1
1
2
3
x
-2
6. (a) 4.24 ◦ F/min (b) −3.09 ◦ F/min
8. (a) 3770 cm3 /min (b) 754 cm2 /min
7. (a) 23 m/s (b) 3.4 m/s (c) At t = 4.75 s.
9. Approximately 2, 0, −1, 0, and 1
10. (a) 5000/hour (b) Approximately 10,000/hour (c) At t = 4 hours (d) At t = 3 hours
11. (a) 12.5 m/s (b) Approximately 10 m/s (c) Approximately 20 m/s
12. (a) Approximately 58 m/s (b) Approximately 10 m/s2 (c) Approximately 14.4 m/s2