Chapter 2

What is a function

A set of ordered pairs

Steps and example for calculating the average rate of change for a function

The meaning of the rate of change of a function

if the average rate of change for f(x), WRT, x, on [a,b] = R. then as x increases from a to b y changes R times as fast as x on average

For example: Average Rate of Change of ln(x²+2x) WRT x over [1,10] =ln40/9 which is estimated to =0.41. As x increases from 1 to 10, y-=ln(x^²+2) increases 0.41 times as fast as x increases on average.

How to mathematically describe 1D motion

An object moves in a straight line. lets call the line of motion the y-axis. (it could be the x-axis). Therefore, the object is moving along a number line. Wherever it is, that point has a coordinate. Since the motion is 1D, the coordinate is one number

Define y=the objects coordinate, that is posistion

t=time

Then y is a function of t: y=f(t). (it coudl also be the functon og g(t), h(t) etx.

The function f(t) contains all information about the objects motion.

How to calculate average speed for 1 Dimensional motion

you literall use the average rate of change function

secant slope between x1 and x1 and slope of the curve at (X1, f(X1)

f(x1+h)-f(x1)/h

What happens when q=p

msecant=f(x1)-f(x1)/x1-x1 =0/0= undefined

what is the tangent line

is at the limiting value of the secant line as Q approaches P

and the slope of the curve is at Point P is the slope of the tangent line at Point P

How do you find the slope of the graph (also called the slope of the tangent line) on a point of the curve

the difference quotient

sandwhich theorom

states that if a function is squeezed between two other functions that have the same limit at a point, then the function must also converge to that limit.

Key Statement of the definition of lim x approaches c of f(x)

If x is close to c [but not = to c] then f(x) is closer to L