1. The Quick Guide to Calculus

2. Take it to the Limit

4. The derivative

5. Derivative
A derivative measures how much a function changes for various inputs of that function. It is like the instantaneous slope at any point on a function (and this can be complex or simple depending on the function)

6. It can help to think about it graphically
What is the slope at every point along the function?
What would that make the derivative look like?

7. What will the derivative look like?
y = a
dy/dx = ??
dy/dx = 0

8. What will the derivative look like?
y = mx
dy/dx = ??
dy/dx = m

9. What will the derivative look like?
y = x2
dy/dx = ??
dy/dx = 2x

10. examples
Can you match the graphs on the left to their derivative functions on the right? 1 a
1 ____
2 ____
3 ____
4 ____ b a d c 2 b 3 c 4 d

11. Now let’s look at it mathematically

12. The “Power Rule”

13. Other Important Rules
But also, from the power rule:

14. Other Important Rules

15. Now YOU try it
Determine the derivatives of the following functions

16. 1. y = x3
4. y = 4
dy/dx = 0
dy/dx = 3x2
2. y = 2x2
5. y =x-4
y’ = 4x
y’ = -4(x)-5
3. y = 3x4 – 8x
6. y = ½ x1/2
d/dx (y) = 12x3 – 8
d/dx (y) =1/4 x-1/2

17. hoseheads?
Heck &

18. Integrals: The ANTI Derivative
An integral is opposite of a derivative
If 2x is the derivative of x2, then x2 is the integral (or anti-derivative) of 2x
What would the integral of of 4x3 be? x4

19. Integral: The Area Under A Curve
The area under a curve can be found by dividing the whole area into tiny rectangles of a finite width and a height equal to the value of the function at the center of each rectangle
This becomes more precise the smaller you make the rectangles
Then you add up all the rectangles

20. Integral: The Area Under a Curve
The approximation to the area becomes better as the rectangles become smaller (N∞,Δx0) and this is what an integral is:

21. Integral: Some examples
For a function that is just a constant, a, then the area under the curve would be a rectangle:
For a linear function f(x)=ax, the area under the curve would be a triangle:

22. Integral: The Anti-Derivative
The general equation for the integral:
Remember that for a derivative it was:
(So the equation for the integral should make sense, it’s the anti-derivative)

23. Now YOU try it
Determine the Integrals of the following functions

24. 1. f(x) = 6x5
4. y = 4
2. f(x) = -6x-7
5. f(x) =½x-½
3. y = 10x4 + x
6. y = ½ x1/2