The Shape of a Graph (4.2)
Increasing/Decreasing Test
(a) If f (x) > 0 on an interval, then f is increasing on that interval.
(b) If f (x) < 0 on an interval, then f is decreasing on that interval.
The First Derivative Test
Suppose that c is a critical po
Maximum and Minimum Values (Extrema) (4.1)
Definitions
Let f be defined on domain D
1.
f (c) is the absolute maximum of f if f (c) f (x) for all x in D.
2.
f (c) is the local maximum of f if f (c) f (x) and c is an interior point of some interval I D.
3.
Properties of the Definite Integral (5.2, Pg. 356)
Let f and g be integrable functions on an interval that contains a, b, and p.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
a
f (x) dx = 0
b
f (x) dx = f (x) dx
a
a
b
a
b
a
b
[ f (x) + g(x)] dx =
a c dx
b
a
b
a
b
a
b
a
b
f
Math 1A
Problem Solving Strategies
Strategy for Solving: Optimization Problems
1.
Identify the quantity to be optimized.
2.
Write a formula involving that quantity.
3.
Express that quantity as a function of a single variable.
4.
Find the optimum value of
Numerical Techniques
Midpoint Rule:
b
a f (x) dx
M n = x f ( x1 ) + f ( x2 ) + f ( xn )
Where x =
Trapezoidal Rule
b
a
ba
and xi = midpoint of [ xi1 , xi ]
n
f (x) dx Tn =
Where x =
x
f ( x0 ) + 2 f ( x1 ) + 2 f ( x2 ) + 2 f ( xn1 ) + f ( xn )
2
ba
a
Differentiation Rules
Derivative of
a constant
d
[c] = 0
dx
Power
rule
d n
x = nx n1
dx
Constant
multiple
d
d
cf (x)] = c [ f (x)]
[
dx
dx
Sum
rule
d
d
d
f (x) + g(x)] = [ f (x)] + [ g(x)]
[
dx
dx
dx
Difference
rule
d
d
d
f (x) g(x)] = [ f (x)] [ g(x)]
[
LIMIT LAWS
Suppose that c is a cOnstant and that hm f (x) and lim g(x) exist.
Then Ha Ha
1. ligmwgon: limf(x)+ 11m g<x> Sum
2. Emagen:gf<x>~hzg<x) , Diffeijence
3. . :1:[cf(x)] ; chm 1:2; 3" I Constant
4! E[f(x)g(x)]:_f(x)~ Ego) , Piiict
s. iim 19:; = 312
Hyperbolic Functions (6.10)
Definitions
e x + e x
Hyperbolic Cosine: cosh x =
2
e x e x
Hyperbolic Sine: sinh x =
2
Graphs
y
y
2
3
2
1
1
1
2
y = sinh x
1
y = cosh x
2
2 1
1
x
1
2
2
An Identity
cosh 2 x sinh 2 x = 1
Two Derivatives
d
[cosh x] = sinh x
dx
d
Table of Indefinite Integrals
k dx
= kx + C
c f (x) dx
= c f (x) dx
x n+1
n
+C
x dx =
n
+
1
1
dx = ln x + C
x
e
ax
x
+C
a dx =
ln a
x
dx = e x + C
sin x dx
= cos x + C
cos x dx
= sin x + C
2
sec
x dx = tan x + C
2
csc
x dx = cot x + C
sec x ta
Horizontal Asymptotes
Definition
If lim f (x) = L or lim f (x) = L , then the line y = L is called a
x
x
horizontal asymptote of the graph of f.
Rational Functions
n(x) an x n + an1 x n1 + a1 x + a0
=
Let f be the rational function: f (x) =
d(x) bm x m +
Olivia Holmes #11
Chris Nelson #
Daniel Mah #
2016
The Golden Ratio
Leonardo Fibonacci backed up Platos belief that the golden ratio was derived
from nature in the twelfth century when the mathematician discovered a recurrent
mathematic sequence that was
MATH 27
Exam #2
Spring 2016
Name: _
Roll #: _ Score: _/100
Work each problem in the space provided on the exam page. Full or partial credit will be given only if all
work has been shown and the answers are clearly indicated. Each numbered problem is worth
Geometry Facts Handout
As you begin your study of trigonometry, you will find it useful to know the
following basic definitions and facts from plane geometry.
I.
Lines and angles
!
!
!
Acute angle
0 < < 90
Right angle
= 90
Obtuse angle
90 < < 180
#
!
"
!
Similar Triangles Handout
When an architect draws plans for a house or an engineer makes a
drawing of a machine part, the result is drawn to scale showing the
same objects in reduced sizes. Two figures with the same shape are
said to be similar.
The silho
MATH 27
Exam #4
Spring 2016
Name: _
Roll #: _ Score: _/100
Work each problem in the space provided on the exam page. Full or partial credit will be given only if all
work has been shown and the answers are clearly indicated. Each numbered problem is worth
MATH 27
Exam #1
Spring 2016
Name: _
Roll #: _ Score: _/100
Work each problem in the space provided on the exam page. Full or partial credit will be given only if all
work has been shown and the answers are clearly indicated. Each numbered problem is worth
MATH 27
Exam #3
Spring 2016
Name: _
Roll #: _ Score: _/100
Work each problem in the space provided on the exam page. Full or partial credit will be given only if all
work has been shown and the answers are clearly indicated. Each numbered problem is worth
MATH 27
Special Right Triangles
There are two special right triangles that frequently arise in mathematics: the 45-45-90 triangle and the
30-60-90 triangle. The sides of each of these triangles have special relationships.
Each angle of a square has a
meas
Fundamental Trigonometric Identities (Pg. 350)
Reciprocal Identities
sin x =
1
csc x
cos x =
1
sec x
tan x =
1
cot x
csc x =
1
sin x
sec x =
1
cos x
cot x =
1
tan x
cot x =
cos x
sin x
Quotient Identities
tan x =
sin x
cos x
Pythagorean Identities
sin 2 x
Angles, Triangles, and Trigonometry
The Unit Circle
Radian Measure
s
r
The radian measure of an angle is defined by the
s
equation = .
r
Note that when is measured in radians, arc length
is given by the equation s = r .
Right Triangle Trig Ratios
Hyp
Opp
Transforming Graphs of Functions (2.4)
Translations
Let c be a positive number. Changes in the graph of y = f (x) are represented as follows.
1.
Vertical shift c units upward:
h(x) = f (x) + c
2.
Vertical shift c units downward:
h(x) = f (x) c
3.
Horizont
Math 5 Lab 1: SAGE and Geometry of Linear Transformations
Due Date: Feb 22, 2017
Lab Goals:
1. learn how to define matrices, vectors, and linear transformations in Sage.
2. learn how to computer reduced row echelon form and matrix algebra (addition, subtr
Melvin: Math 4 k Ev
Exam #1, Mon, Feb 22, 2016 Name: l
Exam Instructions:
-1.
Use scratch paper for scratch work. Write your nal answers and proofs on the
exam in the spaces provided.
. You have 1 hour and 40 minutes to complete this exam.
. You are allow
Elementary row operations
1.(Replacement) Replace one row by the sum of itself and a
multiple of another.
2. (Interchange) Interchange Two Rows
3.(Scaling) Multiply All Entries In A Row By Nonzero Constant
DEFINITION A rectangular matrix is in echelon for
Section 2.1
Theorem 1
Let A, B, and C be matrices of the same size, and let r and s be scalars.
a. A + B = B + A
d. r(A + B) = rA + rB
b. (A + B) + C = A + (B + C) e. (r + s)A = rA + sA
c. A + 0 = A
f. r(sA) = (rs)A
Definition
If A is an mn matrix, and if
Section 4.1 - Vector Spaces and Subspaces
Definition
A vector space is a nonempty set V of objects, called vectors, on
which are defined two operations, called addition and multiplication
by scalars (real numbers), subject to the ten axioms (or rules) lis
Elementary row operations
1.(Replacement) Replace one row by the sum of itself and a multiple of another.
2. (Interchange) Interchange Two Rows
3.(Scaling) Multiply All Entries In A Row By Nonzero Constant
DEFINITION A rectangular matrix is in echelon for