**What is the derivative of Sine (2t)**

A. -2 sine (2t)

B. -2 cosine (2t)

C. 2 sine (2t)

D. 2 cosine (2t)

A. -2 sine (2t)

B. -2 cosine (2t)

C. 2 sine (2t)

D. 2 cosine (2t)

**What is the Derivative of**

A bicycles position-vs-time graph is shown below

What is the bicyclist's velocity at t = 10 s, 25 s, & 35 s?

TIME = 10 s

A. -10 m/s

B. -2.5 m/s

C. 0 m/s

D. 2.5 m/s

E. 10 m/s

TIME = 25 s

A. -10 m/s

B. -2.5 m/s

C. 0 m/s

D. 2.5 m/s

E. 10 m/s

TIME = 35 s

A. -10 m/s

B. -2.5 m/s

C. 0 m/s

D. 2.5 m/s

E. 10 m/s

**Constant Acceleration:**

any examples?

All of what we will do with motion

will have constant acceleration

(rare exceptions will be stated)

any examples?

All of what we will do with motion

will have constant acceleration

(rare exceptions will be stated)

Example what can we do with this? (acceleration)

**Now this equation takes the form of:**

**From the definition of acceleration**

we have DERIVED an equation for Velocity.

we have DERIVED an equation for Velocity.

Yeah, but I thought we already had an equation for velocity?

So what is the difference between them

and when would you use them?

In a similar manner we can derive all sorts of kinematics.

Your book does just fine, let me know if you want to go through it, you can come see me later.

Let us rather collect the main three equations of motion

Velocity (no time)

Velocity

Position

Collect the three:

To use these equations I need:

A. Initial position

B. Initial velocity

C. Acceleration

D. Time

E. Some / All : Depends

To solve for 2 (final position) directly

I only need:

1. initial position 2. final position

3. inital velocity 4. final velocity

5. acceleration 6. time

A. 1, 3, & 5

B. 1, 3, & 4

C. 1, 3, 4, & 5

What is Free Fall?

"Can you explain more in depth about derivatives and how they are used?"

"I think I have a good grasp on what instantaneous velocity is, but I get lost while trying to understand how to find the instantaneous velocity. Could you go over this in class?"

INSTANTANEOUS VELOCITY?!!!???!!!

"I know this isn't a calculus class, but can you cover derivatives? "

"I'm having trouble interpreting the velocity and acceleration vs time graphs"

"Can you help me to understand how to find the area under a curve?"

Calculus review

derivative : change in one quantity , while something else is changing

so for velocity...

(while the time changes, how does the______change?

Idea of a limit

we can measure two different positions (and thus calculate displacement)

But it takes time

as the time gets smaller we do better at getting the Instantaneous velocity