Lecture Notes 7
Rates of Change and the Derivative
Rates of Change and the Derivative
Introduction to Computational Science , Fall 2010
Marcus Pendergrass
Hampden-Sydney College
Example: Height Versus Time
Lecture Notes 7
Dr. Pendergrass
I
A ball is tossed straight up from a bridge that is 11 meters
above the water.
I
Newton’s laws give us a model for the height of the ball as a
function of time:
s(t) = 11 + 15t − 4.9t2
I
s = height in meters above water
I
t = time in seconds after release
Dr. Pendergrass (H-SC)
Lecture Notes 7
Fall 2010
2/3
Example: Height Versus Time
Lecture Notes 7
Dr. Pendergrass
I
A ball is tossed straight up from a bridge that is 11 meters
above the water.
I
Newton’s laws give us a model for the height of the ball as a
function of time:
s(t) = 11 + 15t − 4.9t2
I
s = height in meters above water
I
t = time in seconds after release
Dr. Pendergrass (H-SC)
Lecture Notes 7
Fall 2010
2/3
Example: Height Versus Time
Lecture Notes 7
Dr. Pendergrass
I
A ball is tossed straight up from a bridge that is 11 meters
above the water.
I
Newton’s laws give us a model for the height of the ball as a
function of time:
s(t) = 11 + 15t − 4.9t2
I
s = height in meters above water
I
t = time in seconds after release
Dr. Pendergrass (H-SC)
Lecture Notes 7
Fall 2010
2/3
Example: Height Versus Time
Lecture Notes 7
Dr. Pendergrass
I
A ball is tossed straight up from a bridge that is 11 meters
above the water.
I
Newton’s laws give us a model for the height of the ball as a
function of time:
s(t) = 11 + 15t − 4.9t2
I
s = height in meters above water
I
t = time in seconds after release
Dr. Pendergrass (H-SC)
Lecture Notes 7
Fall 2010
2/3
Example: Height Versus Time
Lecture Notes 7
Dr. Pendergrass
I
A ball is tossed straight up from a bridge that is 11 meters
above the water.
I
Newton’s laws give us a model for the height of the ball as a
function of time:
s(t) = 11 + 15t − 4.9t2
I
s = height in meters above water
I
t = time in seconds after release
Dr. Pendergrass (H-SC)
Lecture Notes 7
Fall 2010
2/3
Example: Height Versus Time
Lecture Notes 7
s(t) = 11 + 15t − 4.9t2
t (sec)
s(t) (m)
t (sec)
s(t) (m)
0.00
11.0000
2.00
21.4000
Dr. Pendergrass (H-SC)
0.25
14.4438
2.25
19.9437
Dr. Pendergrass
0.50
17.2750
2.50
17.8750
0.75
19.4938
2.75
15.1937
1.00
21.1000
3.00
11.9000
Lecture Notes 7
1.25
22.0938
1.50
22.4750
1.75
22.2438
3.25
7.9938
3.50
3.4750
3.75
-1.6563
Fall 2010
3/3
Example: Height Versus Time
Lecture Notes 7
s(t) = 11 + 15t − 4.9t2
t (sec)
s(t) (m)
t (sec)
s(t) (m)
0.00
11.0000
2.00
21.4000
Dr. Pendergrass (H-SC)
0.25
14.4438
2.25
19.9437
Dr. Pendergrass
0.50
17.2750
2.50
17.8750
0.75
19.4938
2.75
15.1937
1.00
21.1000
3.00
11.9000
Lecture Notes 7
1.25
22.0938
1.50
22.4750
1.75
22.2438
3.25
7.9938
3.50
3.4750
3.75
-1.6563
Fall 2010
3/3
Example: Height Versus Time
Lecture Notes 7
s(t) = 11 + 15t − 4.9t2
t (sec)
s(t) (m)
t (sec)
s(t) (m)
0.00
11.0000
2.00
21.4000
Dr. Pendergrass (H-SC)
0.25
14.4438
2.25
19.9437
Dr. Pendergrass
0.50
17.2750
2.50
17.8750
0.75
19.4938
2.75
15.1937
1.00
21.1000
3.00
11.9000
Lecture Notes 7
1.25
22.0938
1.50
22.4750
1.75
22.2438
3.25
7.9938
3.50
3.4750
3.75
-1.6563
Fall 2010
3/3
Example: Height Versus Time
Lecture Notes 7
s(t) = 11 + 15t − 4.9t2
t (sec)
s(t) (m)
t (sec)
s(t) (m)
I
0.00
11.0000
2.00
21.4000
0.25
14.4438
2.25
19.9437
Dr. Pendergrass
0.50
17.2750
2.50
17.8750
0.75
19.4938
2.75
15.1937
1.00
21.1000
3.00
11.9000
1.25
22.0938
1.50
22.4750
1.75
22.2438
3.25
7.9938
3.50
3.4750
3.75
-1.6563
What is the velocity of the ball at a given time t?
Dr. Pendergrass (H-SC)
Lecture Notes 7
Fall 2010
3/3