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Piecewise Function

Volume of a sphere

Horizontal Line Test

Vertical Line Test

Ex. y ={-3x - .5, -1 < x < 1 ; 2x + 1, 1 <= x <= 3

4/3 ** r^3 ** PI

A test to see if a horizontal line passes through the graph m…

A test use to determine if a relation is a function. A relati…

Piecewise Function

Ex. y ={-3x - .5, -1 < x < 1 ; 2x + 1, 1 <= x <= 3

Volume of a sphere

4/3 ** r^3 ** PI

Definition of Continuity

Intermediate Value Theorem

Definition of Derivative

Alt. Form of Derivative

lim f(x) x->c f(x) = f(c)

1. If f is continuous [a,b] 2. and f(a) < k < f(b) or f(b) <…

f'(x) = lim h-> 0 f(x+h) - f(x) / h

f'(x) = lim x-> c f(x) - f(c) / x-c

Definition of Continuity

lim f(x) x->c f(x) = f(c)

Intermediate Value Theorem

1. If f is continuous [a,b] 2. and f(a) < k < f(b) or f(b) <…

Corner

Vertical tangents

Cusp

Discontinuity

Piece wise and absolute value functions

_... 1/odd or odd\| Infinity on both sides

Even/odd power<1 one side +infinity one side -infinity

0 in denominator

Corner

Piece wise and absolute value functions

Vertical tangents

_... 1/odd or odd\| Infinity on both sides

Telescoping series

Harmonic series

Geometric series

Integral test

1/(n * (n+1))... -converges at 1

1 + 1/2 + 1/3 + 1/4 +...... -diverges

a + ar + ar^2 + ar^3 +...... -converges with sum S = a/(1 - r) I…

series E(an) where f(n) = an and f(x) is when x replaces n. I…

Telescoping series

1/(n * (n+1))... -converges at 1

Harmonic series

1 + 1/2 + 1/3 + 1/4 +...... -diverges

Infinite Series

Infinite Series (sequence of partial s…

Convergence of an Infinite Series

Geometric Series

Sn =a1+a2+a3....+an

S1 =a1 ... S2= a1+ a2... Sn = a1+a2+....+an

In inifinite series converges to S if lin Sn as N approaches…

a + ar + ar^2 = ar^3 +....> + ar^n-1 + ....

Infinite Series

Sn =a1+a2+a3....+an

Infinite Series (sequence of partial s…

S1 =a1 ... S2= a1+ a2... Sn = a1+a2+....+an

Two Special Trigonometric Limits

Definition of the Derivative of a Func…

Average Velocity or Average rate of Ch…

Properties of the Natural Logarithmic…

limx-->0 (sinx) / x = 1... lim x-->0 (1-cosx) / x = 0

f ' (x) = lim Δx-->0 [f(x+Δ x) - f(x)] / Δx

[f(b) - f(a)] / (b - a)

a. The domain is (0, ∝) and the range is (-∝,∝)... b. ln(1) = 0…

Two Special Trigonometric Limits

limx-->0 (sinx) / x = 1... lim x-->0 (1-cosx) / x = 0

Definition of the Derivative of a Func…

f ' (x) = lim Δx-->0 [f(x+Δ x) - f(x)] / Δx

Function

Relation

Domain

Range

A relationship from one set (called the domain) to another se…

An association between two or more variables. If one of the v…

All of the input or x values in a function

All of the output or y values in a function

Function

A relationship from one set (called the domain) to another se…

Relation

An association between two or more variables. If one of the v…

10

5

6

3/2

Evaluate the limit in Part A given the following information:

Evaluate the limit in Part B given the following information:

Evaluate the limit in Part C given the following information:

Evaluate the limit in Part D given the following information:

10

Evaluate the limit in Part A given the following information:

5

Evaluate the limit in Part B given the following information:

Complex Fraction

Compound Inequality

Cube root

Distance Formula

A fraction that contains one or more fractions in its numerat…

Two or more inequalities joined together by "and" or "or"

a number that when multiplied three times equals a given number

d = √[( x₂ - x₁)² + (y₂ - y₁)²]

Complex Fraction

A fraction that contains one or more fractions in its numerat…

Compound Inequality

Two or more inequalities joined together by "and" or "or"

10

5

6

3/2

Evaluate the limit in Part A given the following information:

Evaluate the limit in Part B given the following information:

Evaluate the limit in Part C given the following information:

Evaluate the limit in Part D given the following information:

10

Evaluate the limit in Part A given the following information:

5

Evaluate the limit in Part B given the following information:

10

5

6

3/2

Evaluate the limit in Part A given the following information:

Evaluate the limit in Part B given the following information:

Evaluate the limit in Part C given the following information:

Evaluate the limit in Part D given the following information:

10

Evaluate the limit in Part A given the following information:

5

Evaluate the limit in Part B given the following information:

Fundamental Theorem of Algebra

Complex Zeros Occur in Conjugate Pairs

Logarithmic Functions

Natural Logarithmic Functions

1.A polynomial of degree n, where n>o, has exactly n roots in…

1.If a + bi, where b does not equal 0, is a zero of the funct…

1.For x>0, a>0, and a can't equal 1, y=log of x with base a i…

1.For X>0, y=lnx if and only if X=e to the y power.... 2.The fun…

Fundamental Theorem of Algebra

1.A polynomial of degree n, where n>o, has exactly n roots in…

Complex Zeros Occur in Conjugate Pairs

1.If a + bi, where b does not equal 0, is a zero of the funct…

Derivative of a function at a point

Mean Value Theorem

Critical point of a function

Local minimum

Limit of the slopes of secant lines to provide the slope of t…

Slope of secant joining the endpoints of an interval from a t…

A number in the domain of f where either the derivative equal…

Local minimum occurs at a critical point where the sign of th…

Derivative of a function at a point

Limit of the slopes of secant lines to provide the slope of t…

Mean Value Theorem

Slope of secant joining the endpoints of an interval from a t…

Function

Relation

Domain

Range

A relationship from one set (called the domain) to another se…

An association between two or more variables. If one of the v…

All of the input or x values in a function

All of the output or y values in a function

Function

A relationship from one set (called the domain) to another se…

Relation

An association between two or more variables. If one of the v…

Find the zeros

show that f(x) is even

show that f(x) is odd

show that lim(x→a) of f(x) exists

Find roots... set function=0... factor or use quad equation... graph t…

show that f(-x)=f(x)... symmetric to y axis

show that f(-x)= -f(x) OR f(x)= -f(-x)... symmetric around origin

show that lim(x→a⁻) f(x)= lim (x→a⁺) f(x) exists and are equal

Find the zeros

Find roots... set function=0... factor or use quad equation... graph t…

show that f(x) is even

show that f(-x)=f(x)... symmetric to y axis

Intermediate Value Theorem

If f is continuous on [a,b] and k is b…

Rolle's Theorem

If f is continuous on [a,b] and f is d…

If f is continuous on [a,b] and k is between f(a) and f(b), t…

there is a number c in [a,b] such that f(c)=k.... (Intermediate…

If f is continuous on [a,b] and f is differentiable on (a,b)…

there is at least one number c in (a,b) such that f'(c)=0.... (…

Intermediate Value Theorem

If f is continuous on [a,b] and k is between f(a) and f(b), t…

If f is continuous on [a,b] and k is b…

there is a number c in [a,b] such that f(c)=k.... (Intermediate…

Two Special Trigonometric Limits

Definition of the Derivative of a Func…

Average Velocity or Average rate of Ch…

Guidelines for Implicit Differentiation

limx-->0 (sinx) / x = 1... lim x-->0 (1-cosx) / x = 0

f ' (x) = lim Δx-->0 [f(x+Δ x) - f(x)] / Δx

[f(b) - f(a)] / (b - a)

a. Differentiate both sides of the equation with respect to x…

Two Special Trigonometric Limits

limx-->0 (sinx) / x = 1... lim x-->0 (1-cosx) / x = 0

Definition of the Derivative of a Func…

f ' (x) = lim Δx-->0 [f(x+Δ x) - f(x)] / Δx

linear equation, three forms- what are…

slope-intercept form:

point-slope form:

standard form:

slope-intercept form, point-slope form, and standard form

y = mx + b

y-y₁=m(x-x₁)

Ax + By = C

linear equation, three forms- what are…

slope-intercept form, point-slope form, and standard form

slope-intercept form:

y = mx + b