Basic Differentiation: Power, Product, Quotient and Chain Rules
1. Find f ( x ): f(x) = x2 + 3 sin x.
2. Find f ( x ): f(x) = x2 + 2 cos x.
: y = 6 sin x + 4 cos x 3.
4. Find the instantaneous rate of change of R with respect
Basic Differentiation Rules Test - Make Up
INSTRUCTIONS: Show all of your work underneath each question or neatly on your answer sheet.
1. Differentiate: y =
3x 2 3
(1 + x 2 ) 3
3(1 x 2 )
(1 + x 2 ) 2
DEFINITION OF INSTANTANEOUS RATE OF CHANGE
Graphically: The instantaneous rate of change is the slope of the tangent line to the
curve at a specific point.
Algebraically: Instantaneous Rate of Change at x = a is
f a h f a
: The instantaneous
DEFINITION OF AVERAGE RATE OF CHANGE
Graphically: The average rate of change between two points is the slope of the line
passing through those two points on a graph. This line is called a secant line.
Algebraically: Average Rate of Change =
Total Change i
TYPES OF DISCONTINUITIES
There are 2 types of discontinuities
Type 1: Removable. A
occurs when there is a
hole in the graph.
Type 2: Non-Removable. A nonremovable disconuity
occurs when there is a vertical asymptote or a jump.
INTERMEDIATE VALUE THEOREM
If f is continuous on the closed interval [a, b] then f takes on every value between f (a)
and f (b).
Suppose k is any number between f(a) and f(b), then there is at least one number c in
[a, b] such that f(c) = k.
: The Interme
DEFINITION OF CONTINUITY AT x = c
A function y = f (x) is continuous at x = c if
lim f ( x ) = f (c ) .
: This last statement implies 2 things:
1) lim f ( x ) exists
2) f (c) exists
Keep in mind The limit in part (1) exists if and only if lim f ( x
FINDING HORIZONTAL ASYMPTOTES
Three possibilities exist when finding the y-value that a function approaches as x :
1) The numerator grows faster NO H.A.
2 x3 3x 1
DNE b/c the numerator grows faster than the denominator,
3x 2 7
DEFINITION OF A VERTICAL ASYMPTOTE
The line x = a is a vertical asymptote of the graph of a function y = f (x) if either
lim f ( x ) =
lim f ( x ) =
Important : Infinity is NOT a number, and thus the limit FAILS to exist in both of
WHEN LIMITS FAIL 3 WAYS
The lim f ( x ) does not exist when there is no number satisfying the definition.
1. f (x) approaches a different numbers from the right and left.
x 0 x
2. f (x) increases or decreases without bound as x approach
SLANT "OBLIQUE" ASYMPTOTES
Slanted or Oblique asymptotes occur in rational functions where the degree of the
numerator is higher than the degree of the denominator.
: For functions with oblique asymptotes, lim f x does not exist.
To find the asymptote,
DEFINITION OF A HORIZONTAL ASYMPTOTE
The line y = b is a horizontal asymptote of the graph of a function y = f (x) if either
lim f ( x ) = b
lim f ( x ) = b
In other words if the end behavior of a function (in either direction) approaches a
Chapter 7 Review Test
1. A survey conducted by Black Flag asked whether or not the action of a certain type of roach disk was
effective in killing roaches. 79% of the respondents agreed that the roach disk was effective. The number
79% is a