Youth Curriculum

1. DERIVATIVES,
DERIVATIVES,
DERIVATIVES

2. QUIZ TIME
• If x means distance
• If v means speed
dx
dt
= ?
dv
dt
= ?

3. REVIEW
• If you have a function for position…
• You can take a ________ of position to get ________.
• If you have speed, you can take the ________ to get ________.

4. THE CONNECTIONS
x Position
dx
dt
Speed
dv
dt
Acceleration

5. THE ARNOLD SCHWARZENEGGER OF MATH
LEONARD EULER
• Euler was a Swiss mathematician, physicist,
astronomer, geographer, logician and engineer.
• Euler became blind. But that didn’t stop him.
He had scribes to write down his math
equations.
• Euler worked in almost all areas of
mathematics, such as geometry, infinitesimal
calculus, trigonometry, algebra, and number
theory, as well as continuum physics, lunar
theory and other areas of physics.
• He produced over 30,000 pages of math,
science, music, and so much more!

6. THE MOST REMARKABLE FORMULA IN MATHEMATICS
EULER’S FORMULA
eiθ
= cosθ + isinθ
x
∂f
∂x
+ y
∂f
∂y
= rf(x, y)
∂f
∂u
du
dλ
+
∂f
∂u
du
dλ
= rλr−1
f(x, y)
• Euler’s identity is used when
things oscillate or move up and
down at any pace:
• This means that this equation is
used in
-Classical Mechanics
-Electromagnetic Theory
-Thermodynamics
-Quantum Mechanics
-Astrophysics
-Engineering
-Chemistry
…Do you recognize
anything in these
equations?

7. BIGGER THAN FLOSSING A FEW YEARS AGO!
EULER WAS AMAZING

8. Example 1:
x = t2
DERIVATIVES
x Position
dx
dt
Speed
dv
dt
Acceleration
Let’s look at how we can actually take a derivative!
REMEMBER BOTH SIDES OF THE
EQUATION REPRESENT POSITION

9. LET’S START WITH POSITION
Example 1:
x = t2t x
0 0
1 1
2 4
3 9
4 16
5 25
6 36
7 49
8 64
9 81
10 100
You guys have to tell me
where to put the dots!
The position function can fill in the gaps between the dots!

10. NOW LET’S TAKE THE DERIVATIVE OF THE POSITION
Example 1:
x = t2
x = 2t
So how did we take a derivative?

11. Example 1:
x = t2
NOW LET’S TAKE THE DERIVATIVE OF THE POSITION
1. Copy the 2 from the top
and put it in front of the t.
2. Subtract 1 from the 2 up
top
3. Sit back and realize that
you are a calculus genius!
x = 2t
x = 2t2
1.
x = 2t1
2.
EINSTEIN GOT NOTHIN’
ON YOU!
3.

12. HOW TO WRITE A DERIVATIVE
d
dt
(t2
) = 2t

13. MORE EXAMPLES OF DERIVATIVES
d
dt
(t3
) = ?
d
dt
(t3
) = 3t2
d
dt
(t4
) = ?
d
dt
(t4
) = 4t3
d
dt
(t30
) = ?
d
dt
(t30
) = 30t29

14. d
dt
(t3
) = 3t2
d
dt
(t4
) = 4t3
d
dt
(t30
) = 30t29
So what are this derivatives. Remember, we were taking the derivative of
POSITION!
Speed = 3t2
Speed = 4t3
Speed = 30t29
v = 3t2
v = 4t3
v = 30t29

15. YAY!
• Congratulations! You took the derivative of a position function t
get speed.
• Ummm… you are a physicist/mathematician now!

16. WHAT IF WE TOOK A DERIVATIVE OF A SPEED FUNCTION
d
dt
(t3
) = 3t2
d
dt
(t4
) = 4t3
d
dt
(t30
) = 30t29
acceleration = 3t2
acceleration = 4t3
acceleration = 30t29
a = 3t2
a = 4t3
a = 30t29