Post on 28-Sep-2020
Part 1
3.4: The Derivative as a Rate of Change
Part 1: Motion Along a Line
MATH 165: Calculus I
Department of Mathematics
Iowa State University
Paul J. Barloon
MATH 165 Section 3.4
Part 1 Motion Along a Line
Classic Example (Yet Again)
Let’s look once again atthe stoplight problem.
The graph actually showsthe car’s positionaccording to the positionfunction s(t) = 2t2.
5 10 15 20 25 30
300
600
900
1,200
1,500
1,800 (30,1800)
(10,200)
(0,0)
time(sec)
distance(feet)
MATH 165 Section 3.4
Part 1 Motion Along a Line
Classic Example (Yet Again)
Let’s look once again atthe stoplight problem.
The graph actually showsthe car’s positionaccording to the positionfunction s(t) = 2t2.
5 10 15 20 25 30
300
600
900
1,200
1,500
1,800 (30,1800)
(10,200)
(0,0)
time(sec)
distance(feet)
MATH 165 Section 3.4
Part 1 Motion Along a Line
Displacement
We found the averagevelocity over the first 30seconds by first finding thedisplacement of the carover that period:
1800 = s(0 + 30)� s(0)
In general, displacementover a time period �t isgiven by:
�s = s(t +�t)� s(t)
5 10 15 20 25 30
300
600
900
1,200
1,500
1,800 (30,1800)
(10,200)
(0,0)
time(sec)
distance(feet)
MATH 165 Section 3.4
Part 1 Motion Along a Line
Displacement
We found the averagevelocity over the first 30seconds by first finding thedisplacement of the carover that period:
1800 = s(0 + 30)� s(0)
In general, displacementover a time period �t isgiven by:
�s = s(t +�t)� s(t)
5 10 15 20 25 30
300
600
900
1,200
1,500
1,800 (30,1800)
(10,200)
(0,0)
time(sec)
distance(feet)
MATH 165 Section 3.4
Part 1 Motion Along a Line
Displacement
We found the averagevelocity over the first 30seconds by first finding thedisplacement of the carover that period:
1800 = s(0 + 30)� s(0)
In general, displacementover a time period �t isgiven by:
�s = s(t +�t)� s(t)
5 10 15 20 25 30
300
600
900
1,200
1,500
1,800 (30,1800)
(10,200)
(0,0)
time(sec)
distance(feet)
MATH 165 Section 3.4
Part 1 Motion Along a Line
Average Velocity
Then, the average velocitywas:
1800
30=
s(0 + 30)� s(0)
30� 0
In general,average velocity over aperiod �t is given by:
�s
�t=
s(t +�t)� s(t)
�t
5 10 15 20 25 30
300
600
900
1,200
1,500
1,800 (30,1800)
(10,200)
(0,0)
time(sec)
distance(feet)
MATH 165 Section 3.4
Part 1 Motion Along a Line
Average Velocity
Then, the average velocitywas:
1800
30=
s(0 + 30)� s(0)
30� 0
In general,average velocity over aperiod �t is given by:
�s
�t=
s(t +�t)� s(t)
�t
5 10 15 20 25 30
300
600
900
1,200
1,500
1,800 (30,1800)
(10,200)
(0,0)
time(sec)
distance(feet)
MATH 165 Section 3.4
Part 1 Motion Along a Line
Average Velocity
Then, the average velocitywas:
1800
30=
s(0 + 30)� s(0)
30� 0
In general,average velocity over aperiod �t is given by:
�s
�t=
s(t +�t)� s(t)
�t
5 10 15 20 25 30
300
600
900
1,200
1,500
1,800 (30,1800)
(10,200)
(0,0)
time(sec)
distance(feet)
MATH 165 Section 3.4
Part 1 Motion Along a Line
Velocity
We then let �t “get small” to find the velocity function for thecar.
Definition
If a body’s position at time t is given by s = f (t), then the body’svelocity (or instantaneous velocity) at time t is:
v(t) =ds
dt= lim
�t!0
f (t +�t)� f (t)
�t
MATH 165 Section 3.4
Part 1 Motion Along a Line
EXAMPLE 1a: Suppose that a body’s position along a straightline is given by the function s(t) = t2 � 3t + 2 for 0 t 2.
What is the body’s displacement over the given interval?
MATH 165 Section 3.4
Part 1 Motion Along a Line
EXAMPLE 1b: Suppose that a body’s position along a straightline is given by the function s(t) = t2 � 3t + 2 for 0 t 2.
What is the body’s average velocity over the given interval?
MATH 165 Section 3.4
Part 1 Motion Along a Line
EXAMPLE 1c: Suppose that a body’s position along a straightline is given by the function s(t) = t2 � 3t + 2 for 0 t 2.
What is the body’s instantaneous velocity at t = 1 second?
MATH 165 Section 3.4
Part 1 Motion Along a Line
EXAMPLE 1d: Suppose that a body’s position along a straightline is given by the function s(t) = t2 � 3t + 2 for 0 t 2.
At what time does the body change direction?
MATH 165 Section 3.4
Part 1 Motion Along a Line
Speed
As we just saw, the sign of a velocity value is meaningful.Sometimes, though, we are interested in the unsigned rate atwhich position is changing.
Definition
Speed is the absolute value of velocity.
Speed = |v(t)| =����ds
dt
����
MATH 165 Section 3.4
Part 1 Motion Along a Line
EXAMPLE 1e: Suppose that a body’s position along a straightline is given by the function s(t) = t2 � 3t + 2 for 0 t 2.
What is the body’s speed at t = 1 second?
MATH 165 Section 3.4
Part 1 Motion Along a Line
Acceleration
The (instantaneous) rate at which velocity changes is a body’sacceleration.
Definition
If a body’s position at time t is given by s = f (t), then the body’sacceleration at time t is:
a(t) =dv
dt=
d2s
dt2
ora(t) = v 0(t) = s 00(t)
MATH 165 Section 3.4
Part 1 Motion Along a Line
EXAMPLE 1f: Suppose that a body’s position along a straightline is given by the function s(t) = t2 � 3t + 2 for 0 t 2.
What is the body’s acceleration at t = 1 second?
MATH 165 Section 3.4
Part 1 Motion Along a Line
Quiz Yourself
The graph at the right show abody’s velocity at time t.
What is the body’s accelerationat time t = 2 seconds?
A) 4
B) 2
C) 0
D) �2
E) �4
MATH 165 Section 3.4
Part 1 Motion Along a Line
The End
MATH 165 Section 3.4
Part 2
3.4: The Derivative as a Rate of Change
Part 2: Related Rates
MATH 165: Calculus I
Department of Mathematics
Iowa State University
Paul J. Barloon
MATH 165 Section 3.4
Part 2 Related Rates
Introduction
In our car (and generic “body”) examples, the independentvariable has always been time (t):
Position = s(t)
Velocity = v(t) = s 0(t)
Acceleration = a(t) = v 0(t) = s 00(t)
We now consider situations in which the independent variable issomething else . . .
MATH 165 Section 3.4
Part 2 Related Rates
Introduction
In our car (and generic “body”) examples, the independentvariable has always been time (t):
Position = s(t)
Velocity = v(t) = s 0(t)
Acceleration = a(t) = v 0(t) = s 00(t)
We now consider situations in which the independent variable issomething else . . .
MATH 165 Section 3.4
Part 2 Related Rates
Introduction
In our car (and generic “body”) examples, the independentvariable has always been time (t):
Position = s(t)
Velocity = v(t) = s 0(t)
Acceleration = a(t) = v 0(t) = s 00(t)
We now consider situations in which the independent variable issomething else . . .
MATH 165 Section 3.4
Part 2 Related Rates
Introduction
In our car (and generic “body”) examples, the independentvariable has always been time (t):
Position = s(t)
Velocity = v(t) = s 0(t)
Acceleration = a(t) = v 0(t) = s 00(t)
We now consider situations in which the independent variable issomething else . . .
MATH 165 Section 3.4
Part 2 Related Rates
Introduction
In our car (and generic “body”) examples, the independentvariable has always been time (t):
Position = s(t)
Velocity = v(t) = s 0(t)
Acceleration = a(t) = v 0(t) = s 00(t)
We now consider situations in which the independent variable issomething else . . .
MATH 165 Section 3.4
Part 2 Related Rates
A Cube
The volume of a cube may beconsidered to be a function ofthe length of the sides l :
V (l) = l3
So, it makes sense to ask aboutthe instantaneous rate of changeof the volume with respect to theside length:
dV
dl=
d
dl[l3] = 3l2
MATH 165 Section 3.4
Part 2 Related Rates
A Cube
The volume of a cube may beconsidered to be a function ofthe length of the sides l :
V (l) = l3
So, it makes sense to ask aboutthe instantaneous rate of changeof the volume with respect to theside length:
dV
dl=
d
dl[l3] = 3l2
MATH 165 Section 3.4
Part 2 Related Rates
EXAMPLE 1a: At what ratedoes the volume of a cubechange with respect to sidelength when l = 5 ft?
MATH 165 Section 3.4
Part 2 Related Rates
EXAMPLE 1b: Byapproximately how much doesthe volume increase when theside length changes from 5 to 5.1feet?
MATH 165 Section 3.4
Part 2 Related Rates
Note that this isjust the slope of thetangent at l = 5when volume isgraphed on thevertical axis andside length on thehorizontal:
MATH 165 Section 3.4
Part 2 Related Rates
Quiz Yourself
Suppose that a spill is forming a circular oil slick around a leakingtanker. How fast does the area of the slick change with respect toits radius when that radius is 90 feet?
A) 45⇡ ft2/ft
B) 90⇡ ft2/ft
C) 180⇡ ft2/ft
D) 810⇡ ft2/ft
E) 8100⇡ ft2/ft
MATH 165 Section 3.4
Part 2 Related Rates
The End
MATH 165 Section 3.4
MATH 165 14–19,32–37 Warm-Up Question – Sep. 14, 2018
Suppose that a body’s position along a straight line
is given by the function s(t) = t2 � 3t + 2 for
0 t 2 (t in seconds).
At what time(s) does the body change direction?
A) t = 0 secs
B) t = 1 sec
C) t = 32 secs
D) t = 2 secs
E) t = 1 and t = 2 secs
F) The body does not change direction
MATH 165 14–19,32–37 Warm-Up Question – Sep. 14, 2018
Suppose that a body’s position along a straight line
is given by the function s(t) = t2 � 3t + 2 for
0 t 2 (t in seconds).
At what time(s) does the body change direction?
A) t = 0 secs
B) t = 1 sec
*C) t = 32 secs
D) t = 2 secs
E) t = 1 and t = 2 secs
F) The body does not change direction