M ath 1016: Elementary Calculus with Trigonometry I
C h. 4 T echniques of Differentiation with Applications
S ec. 4 . 3 : T he Product and Quotient Rules
I.
The Product Rule
A. The Product Rule:
f and g are both differentiable at x, then so is their produ
M ath 1016: Elementary Calculus with Trigonometry I
C h. 3 Introduction to the Derivative
S ec. 3. 2 : Limits a nd Continuity
I.
D efinitions
A. D ef n : A function
if
lim f ( x ) = f (c) .
xc
B. D ef n : A function
if
y = f ( x ) is continuous at a left
Math 1016: Elementary Calculus with Trigonometry I
C h. 3 Introduction to the Derivative
S ec. 3.4: Average Rate of Change
I.
Average Rate of Change
A. Secant Line
1 . Defn: The secant line to the curve y=f(x) is the line connecting two points,
(a, f (a)
M ath 1016: Elementary Calculus with Trigonometry I
C h. 3 Introduction to the Derivative
S ec. 3.5: Derivatives: Numerical and Graphical Viewpoints
I.
Review
A. The average rate of change of
y = f ( x ) wrt x over the interval [ x, x + h ] is
f ( x + h)
M ath 1016: Elementary Calculus with Trigonometry I
C h. 4 T echniques of Differentiation with Applications
S ec. 4 . 1 : Derivatives o f Powers, Sums, & Constant Multiples
I.
Derivative Rules
A. Derivative of a Constant Function
If c is a constant, then
M ath 1016: Elementary Calculus with Trigonometry I
C h. 3 Introduction to the Derivative
S ec. 3. 6 : D erivatives: A lgebraic Viewpoint
I.
Review
f ( a+h )-f ( a )
h
1. average rate of change of f wrt x over [ a, a + h ]
2. Graphically: slope of secant
M ath 1016: Elementary Calculus with Trigonometry I
C h. 3 Introduction to the Derivative
S ec. 3.3: Limits: Algebraic Viewpoints
I.
The Limit Laws
A. Limit Laws
If L, M, c and
and
k are real numbers and let f and g be any 2 functions st lim f ( x ) = L
x
M ath 1 016 : E lementary C alculus w ith Trig onometry I
C h. 3 I ntroduction to the Derivative
S ec. 3 . 1 : L imits : Numerical and Graphical Viewpoints
I.
I ntroduction
A. What does the word limit mean?
Websters Dictionary gives a few different meanin
M ath 1 016 : E lementary Calculus with Trigonometry I
S ec. 1 . 3 : L inear Functions a nd M odels
I.
Linear Function
A. Definitions
1. A linear function is one that can be written in the form
or
f ( x ) = mx + b (function form)
y = mx + b (equation form
M ath 1 016 : E lementary C alculus w ith Trig onometry I
C h. 1 F unctions & Applications
S ec. 1 . 1 : F unctions from the Numerical, Algebraic and Graphical Viewpoints
= for all
= there exists
Def = definition
Thm = Theorem
soln = solution
= perpend
M ath 1 016 : E lementary C alculus w ith Trig onometry I
C h. 2 N onlinear F unctions & M odels
S ec. 2 . 1 : Q uadratic Functions & Models
I.
Quadratic Functions
x is a function that can be written in the
2
form f ( x ) = ax + bx + c (function form) or
M ath 1016: Elementary Calculus with Trigonometry I
S ec. 2.2: Exponential Functions & Models
I.
Review of Exponential Rules
b x b y = b x+ y
(b x ) y = b xy
bx
= b x y
y
b
(bc) x = b x c x
1
= b x
bx
bx
b
=x
c
c
x
b0 = 1
II.
Exponential Functions
x is
M ath 1016: Elementary Calculus with Trigonometry I
T rigonometric Functions and Graphs
I.
Trig Functions
A. Trig Functions and the Unit Circle
Choose
P ( x, y ) a point on the unit circle where the
terminal side of intersects with the circle.
Then cos =
M ath 1016: Elementary Calculus with Trigonometry I
S ec. 2. 3 : L ogarithmic F unctions & Models
I.
Logarithmic Functions
A. Relationship Between Logarithmic Functions and Exponential Functions
1.
y = log b ( x ) means x = b y for x > 0 , b > 0 , b 1
2.