1. The Amazing Power of the Derivative in
Velocity Functions
Calculus
- Introduce topic
Michelle Martin
Period 4
AP #26

2. A particle moves along the x-axis with acceleration given by
for all At , the velocity v(t) of the particle is 2 and the position x(t) is 5.
A. Write an expression for the velocity v(t) of the particle.
B. Write an expression for the position x(t).
C. For what values of t is the particle moving to the
right? Justify your answer.
- Read problem to be solved.
D. Find the total distance traveled by the particle
from t = 0 to

3. Solving the Problem • Numerically • Graphically • Analytically
- Restate slide.

4. Writing the Velocity Function
Part A
Writing the Velocity Function
Given:
v(0)=2
Knowledge:
Solution:
- Introduce the given information.
- Explain the graph of acceleration.
- Tie in the knowledge of how to get a velocity function from the acceleration
function.
- Explain solution step-by-step.

5. Writing the Position Function
Part B
Writing the Position Function
Given:
x(0)=5
Knowledge:
Solution:
- Introduce given information and explain graph of velocity.
- Position function is the antiderivative of the velocity function.
- Explain solution step-by-step.

6. Studying Direction of Movement
Part C
Studying Direction of Movement
Knowledge:
Solution:
- State the velocity function.
- Explain analytical method to find critical values.
- Explain sign study.
v(t) +
No critical values
The particle is always moving to the right.

7. Finding Total Distance Traveled
Part D
Finding Total Distance Traveled
Knowledge:
Distance traveled = Area under velocity graph
- Read slide.
What is the integral of "one over cabin" with respect to "cabin"? Answer: Natural log cabin + c = houseboat.

8. Finding Total Distance Traveled Part Deux
Calculus Overload!
- Explain solving the integral.
- Explain graph of integral of velocity function.
Distance = units

9. The End