**Position, Velocity, & Acceleration**

Regina Brown

**Recall:**

Applications using equations

Just by looking at a given position, velocity, or acceleration graph, it is possible to find the graphs of the non-given functions

**f(x)=position**

f'(x)=velocity

f''(x)=acceleration

f'(x)=velocity

f''(x)=acceleration

Applications using graphs

example:

the position of a car is represented by the function f(x)=3t +5t+2. Find the car's instantaneous velocity and instantaneous acceleration at t=4 minutes.

2

answer:

f'(x)=6t+5

f'(4)=6(4)+5

f'(4)=29 m/s

f"(x)=6

f"(4)=6

f''(4)= 6 m/s

derive f(x) to find the velocity function

plug in 4 for t into f'(x)

solve --> don't forget units

derive f'(x) to find the acceleration funct.

plug in 4 for t into f''(x)

solve --> don't forget units

What you should know how to do

1. Derive/ Integrate to find the unknown functions (P,V, or A)

2. Use graphs to find the unknown functions (P,V, or A)

3. Find instantaneous acceleration & velocity

4. Find distance traveled after a certain amount of time

5. Find average velocity:

6. Find total distance traveled:

**PVA Equations**

**PVA Graphs**

**Sources:**

**What can graphs and their derivatives tell us about P, V, A?**

2

*If you are given just one equation, either derive or integrate to find the other two equations.

Position/ f(x)

Velocity/ f'(x)

Accel./ f''(x)

When given:

Tells us the:

This feature:

slope

concavity

concavity

value

slope

slope

value

value

Of this graph:

value

slope

value

slope

concavity

value

concavity

slope

f'(x)

f'(x)

f"(x)

f(x)

f(x)

f"(x)

f(x)

f'(x)

https://apstudent.collegeboard.org/apcourse/ap-calculus-ab/exam-practice

http://17calculus.com/calc04-linear-motion.php

https://secure-media.collegeboard.org/secure/ap/pdf/calculus-ab/ap-2009-calculus-ab-scoring-guidelines.pdf?__gda__=1387218710_9e52bb14a7e06fc9092d11bbe8a34d3e

f(x)

f'(x)

f"(x)

Derivation

Derivation

Example Problem:

Solution