Section 1.8 Higher Order Derivatives
We will now find the rate of change of the rate of change by differentiating
the derivative:
f (x) or
dy
dx
is the derivative of f(x)
describes the rate of change of y with respect to x
2
f (x) or
dy
dx 2
f(x) or
3
dy
Section R.2
Relations & Functions
Relation set of ordered pairs. The set of xs(inputs) is called the domain and the
set of ys(outputs) is the range
A relation may be specified by:
1. list
cfw_ (1,2) , (2,4) , (3,6) , (4,8) ,
cfw_ (1,2) , (5,6) , (-3,8)
Section R.3
Finding Domain & Range
Representations of sets of real numbers can be expressed as:
I. Set Notation
II. Graph
III. Interval Notation
cfw_x | x > 2
(-, 3]
[
-3
)
7
Brackets indicate endpoint is included while parenthesis indicate it is not inc
Section R.4 Slope & Linear Functions
Important concepts to remember
Slope: m =
rise y y2 y1
run x x 2 x1
m=0
horizontal
Point-Slope form:
, where (x1, y1) and (x2, y2) are points on the line
m is undefined
vertical
m>0
rises to right
m<0
falls to right
y
Section R.5
Types of Functions
A. Polynomial Functions
General polynomial: f(x) = anxn + an-1xn-1 + + a1x1 + a0
of degree n
Degree 1
f(x) = ax + b
Linear Function
Degree 2
f(x) = ax2 + bx + c Quadratic Function (parabola)
If a is positive, the parabola op
Program: Quadratic Formula to be used with TI Calculator (however, it will adapt for other models)
;ClrHome
:Prompt A, B, C
:Disp (-B +
:Disp (-B -
The program screen should look like this
(B2 - 4AC)/(2A)
(B2 - 4AC)/(2A)
1. Push PRGM go over to NEW and pu
Section 1.1 Evaluating Limits Numerically & Graphically
Limit Definition: If f(x) becomes arbitrarily close to a unique number L as x gets
closer to c, then the limit of f(x) as x approaches c is L.
Limit Notation:
lim f ( x) L
xc
read the limit of f(x) a
Section 1.2
Evaluating Limits Algebraically
For any rational or polynomial function, f, with a in the domain then:
lim f ( x) f (a)
x c
1. Try substituting x = a into the function. If the result is a real number,
we have found the limit. Remember , 0 0
c
Section 1.3
Average Rates of Change
Suppose that you drive 300 miles in 6 hours. What is your average velocity? Note
that your average velocity is the average rate of change of distance over time. Notice,
you cannot tell your exact velocity at any specifi
Section 1.4 Differentiation Limits of Difference Quotients
We have learned the difference quotient is a formula for finding:
i)
the slope of the secant line
ii) the average rate of change of the function
We will now use the difference quotient to find the
Section 1.5
Differentiation Techniques
Derivative Notation:
Given y = f(x) , we will use the following notations for its derivative:
1) f (x)
dy
dx
d
f ( x)
3)
dx
dy
4)
dx x 3
2)
read the derivative of f of x or f prime of x
read the derivative of y with
Section 1.6
Product and Quotient Rules
Product Rule
This rule deals with a way of taking the derivative of a product of 2 functions
without actually multiplying first
Suppose
f(x) = g(x) h(x) then
f (x) = g(x) h (x) + h (x) g (x)
or more informally
f (x)
Section 1.7
The Chain Rule
Find the derivative of f(x) = (5x2 3x +4)2 In order to find the derivative,
we must first square the polynomial
(5x2 3x +4) (5x2 3x +4) =
then take the derivative of each term
f (x) =
We will use a more complicated version of th
Section R.1
Graphs and Equations
Precalculus deals with things that are not changing(or changing at a constant
rate). We will tweak a lot of these ideas as we change from the static to the
dynamic world of calculus
Calculus is composed of 2 major branches