The document discusses derivatives and their uses. It begins with an example using a nerf dart gun to explain derivatives. The arch of the nerf dart is a function, and taking the derivative gives the speed. The derivative of speed is acceleration. There are three points where the position does not change: when the dart is not thrown yet, after it lands on the target, and when it briefly pauses at the top of its arc. Derivatives can also be used to find the maximum height of the dart. In summary, derivatives provide the rates of change of functions and can be applied to problems involving distance, speed, and acceleration over time.

1. DERIVATIVES
CONTINUED

2. QUIZ TIME
•
(THE ANSWER IS ONE WORD)
•
(THE ANSWER IS ONE WORD)
d
dt
(Distance) = ?
d
dt
(Speed) = ?

3. GALILEO…THE BOSSMAN
• Before Einstein’s relativity,
• There was the Galilean relativity!
• He also further developed the
kinematic equations
Also known as the equations of
motion

4. GALILEAN TRANSFORMATION
Anyone know what Einstein included?

5. KINEMATIC EQUATIONS
a = (a − r
(
dθ
dt )
2
− r
(
dϕ
dt )
2
sin2
(θ) ̂er . . .
dx
dt
=
dx∘
dt
+
dv
dt
t
x − x∘ =
dx∘
dt
t +
1
2
dv
dt
t2
(
dx
dt )
2
=
(
dx∘
dt )
+ 2
(
dv
dt )
(x − x∘)

6. • The derivative is a measure of the rate of change of a function
• Remember that Speed and Acceleration are both functions!
• Speed is a derivative of distance
• Acceleration is a derivative of Speed
•
•
d
dt
(Distance) = Speed
d
dt
(Speed) = Acceleration
LET’S REVIEW

7. LET’S REVIEW
• The change of position in time is
SPEED
• The change of speed in time is
ACCELERATION
d
dt
(Distance) = Speed
d
dt
(Speed) = Acceleration
How I felt when I learned derivatives!

8. LET’S BREAK THIS DOWN
DERIVATIVES
d
dt
(Distance) = Speed
Take the
derivative
Of
Distance
This gives
you Speed
As your distance changes with time, you get speed
What does it mean to change you distance?

9. LET’S TALK ABOUT AN EXAMPLE
Say Jim Nerf is wanting to shoot a target
with a nerf gun!
However it is kinda a long way away.
How does Jim Nerf aim the nerf gun to hit
the target?
a. To the left
b. To the right
c. Arch it up
d. Arch it towards the floor

10. ARCH IT UPWARDS!!!
THE ANSWER IS C
What does this arch look like? hmmm…
Jim Nerf Here.
Expert with nerf
darts

11. • The arch is a function?
• The arch is very much like
• Since the great nerf dart arch is a
function…
…and the position is changing with
time…
What can we do to a function with a
position that is changing with time?
y = x2
THE GREAT NERF DART ARCH

12. THE ARCH IS A FUNCTION THAT YOU CAN TAKE THE DERIVATIVE OF
What does this arch look like? hmmm…
Jim Nerf Here.
Expert with nerf
darts
You can think of the nerf dart following the arch as the direction of the change

13. LET’S REVIEW
• If you shoot your nerf dart gun upwards it makes a _________
• Since the arch is a ________ , you can take the _________ to
determine the speed.
• If you took the derivative of speed, you get _________.

14. WHAT ELSE ARE DERIVATIVES GOOD FOR?
• Derivatives can be used to determine
the maximum height of the nerf dart
gun!
• A derivative is how the position is
changing.
• There are three points where the
position of the nerf dart’s position does
not change
What would those locations be? (there
are 3)

15. THESE ARE THE THREE PLACES THAT THE FUNCTION DOES NOT CHANGE
Jim Nerf Here.
Expert with nerf
darts
Why are the is the position not
changing at these three points?
Kinda weird….

16. THE ARCH IS A FUNCTION THAT YOU CAN TAKE THE DERIVATIVE OF
Jim Nerf
Here.
Why are the is the
position not
• Remember our definition of a function
Distance and Time
Speed
How fast
you were
going
Input Output
1. The 1st dot is the first place that the
position is not changing.
-Because, Jim nerf has not thrown the
nerf dart yet
2. The dart is the next place that the
position is not changing.
-It already landed and the dart is
stuck to the target.
3. The top dart is there because it has an
split second where the dart ‘hangs’ in
the air.
1 2
3

17. KINEMATIC EQUATIONS
a = (a − r
(
dθ
dt )
2
− r
(
dϕ
dt )
2
sin2
(θ) ̂er . . .
dx
dt
=
dx∘
dt
+
dv
dt
t
x − x∘ =
dx∘
dt
t +
1
2
dv
dt
t2
(
dx
dt )
2
=
(
dx∘
dt )
+ 2
(
dv
dt )
(x − x∘)