1. Calculus for AP physics
When using math to do our work
We’re not a bunch of dopes!
Cuz we know integral means area,
And derivatives is slopes!

2. Pre-Fab Vo cab
FUNCTIONS a relation that uniquely specifies the values of one quantity in terms of some other quantity. Example- for position: r = r(t) Where ‘r’ is the dependent variable and ‘t’ is the independent variable
It is possible for a quantity to depend upon more than one independent variable – we will see some later in the course.
LINEAR FUNCTIONS the independent variable appears as the first power. The equation for the function follows the equation for a straight line: y = mx +b where the slope m tells how the function is changing
NON-LINEAR FUNCTIONS does not have a well defined slope. Derivatives are used to determine the slope over a small segment or the instantaneous slope

3. Pre-Fab Vo cab (cont.)
DERIVATIVE the derivative is a measurement of how a function changes when the values of its inputs change. The derivative of a function at a chosen input value describes the behavior of the function near that input value. The process of finding a derivative is DIFFERENTIATION.
DERIVATIVES ARE SLOPES!!
A derivative is used when the function being analyzed approximates a straight line – this typically occurs when the change in values is very small
Recall the slope of a straight line is determined using DELTA NOTATION (Dy/Dx) since the derivative is over a small interval, the notation changes to:
Which is pronounced
dee why dee eks

4. Pre-Fab Vo cab (cont.)
INTEGRAL in most cases, an integral is an anti-derivative. Integration is used to determine the area under a curve by breaking the entire area into smaller geometric shapes and making a summation.
The Fundamental Theorem of Calculus gives us this definition for integrals: “Suppose the function G(x) is such that its derivative gives F(x):
then

5. STANDARD FORMS – derivatives (Appendix E, p. A-11)

6. EXAMPLES
THE POWER RULE WITH THE SUM/DIFFERENCE RULE (the derivative of a sum/difference is equal to the sum/difference of the derivatives)
Given the function:
Recognize it is the form:
Substitute and solve to obtain:

7. EXAMPLES PRODUCT RULE Substitute and solve to obtain:
Given the function:
Substitute and solve to obtain:
Let: u=(x2+4) and v=(3x3-2x) and
recognize it is in the form:
Check the result by multiplying
the terms and using the power
and sum rules

8. THE QUOTIENT RULE – A SPECIAL CASE
Sometimes it is easier to convert a division problem into a multiplication problem and apply the product rule.
Other times – it is easier to work with the division problem as it is written
The standard form of the QUOTIENT RULE is:

9. EXAMPLE OF THE QUOTIENT RULE
Use the quotient rule to determine the derivative (ds/dt) of the following equation
Let u = t and v =3 – t2
Determine du/dt and dv/dt
Put in standard form
Simplify
Result:

10. THE CHAIN RULE
This is applied when a function contains another function [(x2-4)3 , sin(wt), etc]
Typically the functions are referred to as the outer and inner functions.
The definition of the chain rule is: the derivative of the outer function evaluated at the inner function, all multiplied by the derivative of the inner function
As a formula, the chain rule is:

11. CHAIN RULE EXAMPLE Substitute and solve:
Given the formula: y(x) =(x2-4)3
Let u =(x2-4) and y =(u)3. This makes (x2-4) the inner function and u3 the outer function also, recall the standard form.
Substitute and solve:
Finally: 3(x2-4)2 (2x) = 6x(x2-4)2

12. INTEGRALS INTEGRALS ARE AREAS
Integrals are used to calculate the effects of a quantity when that quantity is not constant
An integral is a limit of a summation
DEFINITE INTEGRALS have limits – INDEFINITE INTEGRALS require the addition of a constant

13. STANDARD FORMS – integrals (Appendix E, p. A-11)

14. EXAMPLE 10t2 + 2t Determine the integral of the following function:
x(t) = 20t + 2
The standard forms for this function are:
10t2 + 2t
Substitute and solve to obtain:

15. INTEGRAL APPLICATION
Determining mass – Consider a rod of length L with a non-uniform mass per unit length l dx
X = 0 dm
(l0 = constant) L
A small piece of the rod (dx) will have a small mass (dm) and the ratio
dm/dx = l.
Adding up all of the pieces will provide the total mass.
integrate and
evaluate

16. STUDY AND PRACTICE!! If you take physics long enough,
You’ll find that math’s a tool.
And if you don’t take physics class-
You’ll just remain a fool!