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Math 124 Final Information Guide
Chapter 1; Library of Functions
Functions and Change
Exponential Functions
New Functions From Old
Logarithmic Functions
Trigonometric Functions
Powers, Polynomials, and Rational Functions7
Chapter 2; The Derivative
2.1 How to we measure speed?
2.2 The Derivative at a Point
2.3 The Derivative Function
2.4 Interpretation of the Derivative
2.5 The Second Derivative
2.6 Differentiability
Chapter 3; Short Cuts to Differentiation
3.1 Powers and Polynomials
3.2 The Exponential Function
3.3 The Product and Quotient Rule
3.4 The Chain Rule
3.5 The Trigonometric Functions
3.6 The Chain Rule and Inverse Functions
3.7 Implicit Functions
3.8 Hyperbolic Functions
3.9 Linear Approximations and the Derivative
Chapter 4; Using the Derivative
4.1 Using the First and Second Derivatives
4.2 Family of Curves
4.3 Optimization
4.4 Applications to Marginality
4.5 Optimization and Modeling
4.6 Rates and Related Rates
4.7 L’Hopital’s Rule, Growth, and Dominance
4.8 Parametric Functions
Chapter 5; The Definite Integral
5.1 How do we Measure Distance Traveled?
5.2 The Definite Integral
5.3 The Fundamental Theorem and Interpretations
5.4 Theorems about Definite Integrals
Chapter 6; Constructing Antiderivatives
6.1 Antiderivatives Graphically and Numerically
6.2 Constructing Antiderivatives Analytically
6.3 Differential Equations
6.4 Second Fundamental Theorem of Calculus
6.5 The Equations of Motion
Chapter 7
7.1 Integration by Substitution
Equations
Chapter 1
Linear Function
y = f(x) = b + mx
y
m=
x
Directly Proportional
y = kx
Exponential Growth; base “a” and “e”
P = P0at Q = Q0et
Exponential Decay: base “a” and “e”
P = P0a-t Q = Q0e-t
Graphing
 Multiplying a function by a constant, c, stretches the graph vertically
(if c > 1) or shrinks the graph vertically (if 0<c<1). A negative sign
(if c<0) reflects the graph about the x-axis, in addition to shrinking or
stretching.
 Replacing y by (y-k) moves a graph up by k (down if k if negative).
 Replacing x by (x-h) moves a graph to the right by h (to the left if h is
negative).
Even Function
f(-x) = f(x) for all x
Odd function
f(-x) = -f(x)
Definition of Inverse function
f-1(y) = x means y = f(x)
Logarithms Properties
Log10x = c means 10c = x
1. loga (uv) = loga u + loga v
1. ln (uv) = ln u + ln v
2. loga (u / v) = loga u – loga v 2. ln (u / v) = ln u – ln v
3. loga un = n loga u
3. ln un = n ln u
Change of Base Formula
m > 0 and b > 0
Arc Length = s = rradiusӨ
Trigonometric Functions and Modifiers
Modified
function
How the graph is obtained
Af(x)
The graph of f with y values scaled by a factor of A
f(x-c)
The graph of f shifted |c| units to the right if c > 0 and to the left if c <
0
f(kx)
The graph of f scaled horizontally be a factor of k
f(x)+b
The graph of f moved vertically b units
The period of
y = Asin(k(x - c)) + d, y = Acos(k(x - c)) + d
where k is a non-zero real number, is 2 /|k|
The amplitude of
y = Asin(k(x - c)) + d, y = Acos(k(x - c)) + d
where k is a non-zero real number, is |A|
Power Functions
f(x) = kxp
Horizontal Asymptote
Let y = f(x) be a function. Suppose that L is a number such that whenever x is large, f(x)
is close to L and suppose that f(x) can be made as close as we want to L by making x
larger.
Then we say that the limit of f(x) as x approaches infinity is L and we write
Vertical Asymptote
Let f be a function which is defined on some open interval containing “a” except possibly
at x = a. We write
if f(x) grows arbitrarily large by choosing x sufficiently close to “a”.
Intermediate Value Theorem: Let f be a function which is continuous on the closed
interval [a, b]. Suppose that d is a real number between f(a) and f(b); then there exists c
in [a, b] such that f(c) = d.
Definition of a limit
The limit of f(x) as x approaches a is L
if and only if, given  > 0, there exists  > 0 such that 0 < |x - a| <  implies
that |f(x) - L| < .
Properties of limits
lim
b = b
xc
lim
x = c
xc
lim
xn = cn
xc
lim
b[f(x)] = b lim f(x)
xc
xc
lim
[f(x) + g(x)] = lim f(x) + lim g(x)
xc
xc
xc
lim
[f(x)g(x)] = [ lim f(x)][ lim g(x)]
xc
xc
xc
lim
f(x) = lim f(x)
xc g(x)
xc
assuming lim g(x)  0
lim g(x)
xc
xc
Definition of Continuity
Chapter 2
Average Velocity =
Change _ in _ Position
Change _ in _ Time
Instantaneous Velocity
Derivative/ Instantaneous Rate of Change
If f’(x) is positive the function is increasing over the interval.
If f’(x) is negative the function is decreasing over the interval
If f’’(x) is positive on an interval then f’(x) is increasing, so the graph of f(x) is concave
up
If f’’(x) is negative on an interval then f’(x) is decreasing, so the graph of f(x) is concave
down
Functions that are not differentiable have/are:
Cusps (sharp corners), piecewise, or oscillate to infinity at a point
Chapter 3
Derivative Short-cuts
(10)
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Rules of differentiation
Sum Rule:
(f+g)'=f'+g'
Constant Multiple Rule:
(cf)'=cf'
Product Rule:
(fg)'=f'g+fg'
Quotient Rule:
Chain Rule:
f(g(x))' = f'(g(x))g'(x)
or
, where y=f(u) and u=g(x)
Derivative of an inverse function
Chapter 4
L’Hopital’s Rule
Parametric Equations for a Straight line
x = x0 + at and y = y0 + at
m = b/a
Parametric- Instantaneous speed
v = [ 
dx  2  dy  2 (1/2)
 +   ]
 dt 
 dt 
Chapter 5
Fundamental Theorem of Calculus
Theorem. Let f be a function which is continuous on the interval [a, b].
Second fundamental theorem of Calculus
(2)
Then
(3)
Integration table of basic functions
1.
2.
3.
; in particular,
4.
5.
7.
6.
Properties of integration
1.
2.
3.
; in particular,
4.
5.
7.
6.
Integration by parts
Or