6
20
10
5
x
0
0.5
1
1.5
2
2.5
0
4
-10
-20
3
-30
0
0.5
1
1.5
2
2.5
x
Graph of altitude, A(t)
Graph of upward velocity, V (t)
Whether the rate the altitude changes –the upward velocity– is
positive or negative is the same as whether the graph itself is rising or falling – in other words, whether the altitude is increasing
or decreasing.
positive rate of change ⇔ graph increasing ⇔ positive slope
negative rate of change ⇔ graph decreasing ⇔ negative slope
February 2, 2005
Sklensky
1
Look at the graphs of P (t) and V (t) in Figure 1 on page 37.
1. Is the derivative of P positive or negative at t = 5? Explain.
2. Is the second derivative of P positive or negative at t = 5?
Explain.
3. Give a value of t where the derivative of P is zero.
February 2, 2005
Sklensky
2
The graph gives the position P (t) of a highway patrol car on the
Mass Pike in miles east of Worcester, where t is minutes after
12:00 noon.
Let V (t) be the car’s velocity at time t.
10
8
6
y
4
2
0
0
2
4
6
8
10
x
1. Where is V (t) positive? negative? zero?
2. When does the car change directions from driving east to
west? from west to east?
3. Use this information to sketch a graph V (t).
4. Where is the second derivative of P positive? negative?
(Use your graph from 3).
5. Sketch a graph of P 00 .
3
February 2, 2005
Sklensky
4