Calculus (Columbia) Flashcards by Brian Higginbotham

Envision
how a
function is like a
machine.

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A
_____ f is a
rule that assigns a
unique value
y = f(x) to each
input x ∈ D?

function

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A
function f is a
_____ that assigns a
unique value
y = f(x) to each
input x ∈ D?

rule

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A
function f is a
rule that assigns a
_____
y = f(x) to each
input x ∈ D?

unique value

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A
function f is a
rule that assigns a
unique value
y = f(x) to each
_____ x ∈ D?

input

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The

*_____** of function f is the
*set** of all
*x ∈ ℝ**
s. t.
*f(x) is defined**.

domain

Picture:
A porn set where a domain-atrix with a red x over her nethers whipping an unsure British Red Guard gimp (who says “∈ℝ…”) while she screams “F(X) IS DEFINED!”

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The

*domain** of function f is the
*_____** of all
*x ∈ ℝ**
s. t.
*f(x) is defined**.

set

Picture:
A porn set where a domain-atrix with a red x over her nethers whipping an unsure British Red Guard gimp (who says “∈ℝ…”) while she screams “F(X) IS DEFINED!”

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The

*domain** of function f is the
*set** of all
*_____**
s. t.
*f(x) is defined**.

x ∈ ℝ

Picture:
A porn set where a domain-atrix with a red x over her nethers whipping an unsure British Red Guard gimp (who says “∈ℝ…”) while she screams “F(X) IS DEFINED!”

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The

*domain** of function f is the
*set** of all
*x ∈ ℝ**
s. t.
*_____**.

f(x) is defined

Picture:
A porn set where a domain-atrix with a red x over her nethers whipping an unsure British Red Guard gimp (who says “∈ℝ…”) while she screams “F(X) IS DEFINED!”

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In set notation, the
domain of function f is
D = {_____ | f(x) is defined}.

x ∈ ℝ

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In set notation, the
domain of function f is
D = {x ∈ ℝ | _____}.

f(x) is defined

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The
domain of
f(x) = x² is
_____.

D = ℝ

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The
domain of
any polynomial is
_____.

D = ℝ

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In set notation,
the
domain of
1
x − 1
is _____.

D = {x ∈ ℝ | x ≠ 1}

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In inverval notation,
the
domain of
1
x − 1
is _____.

D = (−∞, 1) ∪ (1, ∞)

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(a, b)
is an
[open / closed / neither]
interval (or set).

()pen

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[a, b]
is an
[open / closed / neither]
interval (or set).

[ losed

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[a, b)
is an
[open / closed / neither]
interval (or set).

neither

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What is
[a, b]
in
interval notation?

[a, b] = {x ∈ ℝ | a < x < b}

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What is
(a, b)
in
interval notation?

(a, b) = {x ∈ ℝ | a < x < b}

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Envision

A ∩ B.

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Envision

A ∪ B.

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The
_____ of function f is the
set of all
possible values of y = f(x)
for some
x ∈ D.

range

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The
range of function f is the
_____ of all
possible values of y = f(x)
for some
x ∈ D.

set

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The **range** of function *f* is the **set** of **\_\_\_\_\_** of y = f(x) for some **x ∈ D**.

**all possible values**

The **range** of function *f* is the **set** of all **possible values** of y = f(x) for some **\_\_\_\_\_**.

**x ∈ D**

In _set notation_, the **range** of function *f* is R = {**\_\_\_\_\_** | y = f(x) for some x ∈ D}.

**y ∈ ℝ**

In _set notation_, the **range** of function *f* is R = {y ∈ ℝ | **\_\_\_\_\_** for some x ∈ D}.

**y = f(x)**

In _set notation_, the **range** of function *f* is R = {y ∈ ℝ | y = f(x) for some **\_\_\_\_\_**}.

**x ∈ D**

The **domain** of a function is the _[input / output]_ of that function and represents the set of all possible _[independent / dependent]_ variables for that function.

**input** **independent** * INput* * INdependent*

The **range** of a function is the _[input / output]_ of that function and represents the set of all possible _[independent / dependent]_ variables for that function.

**output** **dependent** *"If you're PUT OUT in the cold, you're DEPENDENT on the kindness of others."*

In _interval notation_, what is the **domain** of this function?

**D = [a, b]**Next

In _interval notation_, what is the **range** of this function?

**R = [c, d]**

_Analytically_, what is **Step 1** to determining whether a value is in the **range** of a function?

**Substitute** the value for the dependent variable *(i.e. y = f(x))*

_Analytically_, what is **Step 2** to determining whether a value is in the **range** of a function?

**Solve** for the independent variable. *(i.e. x)*

_After you've done all the work_, how do you know whether a value is in the **range** of a function?

If you can **solve for the independent variable**, then the value is **in the range**.

Given f(x) = _1_ x is * *0** in the * *range** of f(x)?

**No**, because you can't _substitute_ 0 for f(x) and then _solve_. There is no such x-value.

_Graphically_, how can you determine whether a graph is a **function**?

The **vertical line test**. (not rigorous)

How do you know whether a * *graph** * *passes** the vertical line test?

If you **can't draw a vertical line** that **passes through more than one point** on the graph, then it **passes** and is a **function**.

This **fails** the vertical line test, so it's not a function. ## Footnote **What do we have here?**

An **implicit relation** between x and y. (both are independent variables)

A * *\_\_\_\_\_** function has * *different functional forms** over * *different domains**.

**piecewise**

A * *piecewise** function has * *\_\_\_\_\_** over * *different domains**.

**different functional forms**

A * *piecewise** function has * *different functional forms** over * *\_\_\_\_\_**.

**different domains**

How might this **function** be defined?

_Analytically_, polynomials are expressed as p(x) = c₀ + c₁x + c₂x² + ... + c_jxⁿ, * *c_j \_\_\_** and * *n _\>_ 0 is an integer**

**c_j ∈ ℝ**

_Analytically_, polynomials are expressed as p(x) = c₀ + c₁x + c₁x² + ... + c_jxⁿ, * *c_j ∈ ℝ** and * *\_\_\_\_\_**

**n _\>_ 0 is an integer** | (n ∈ ℤ, n _\>_ 0)

How many **variables** can be in a polynomial?

**One**

In a **polynomial**, the **coefficients** can be \_\_\_\_\_.

**any real number**

In a **polynomial**, the **exponents of the variables** can be \_\_\_\_\_.

**any nonnegative integer**

A * *\_\_\_\_\_** is a * *ratio** of * *polynomials**.

**rational function**

A * *rational function** is a * *\_\_\_\_\_** of * *polynomials**.

**ratio**

A * *rational function** is a * *ratio** of * *\_\_\_\_\_**.

**polynomials**

_Analytically_, in an **odd function**, f(−x) = \_\_\_\_\_

**−f(x)**

_Graphically_, an **odd function** will **reflect** about \_\_\_\_\_.

**the y- and x-axes** or the **origin** **O^dd** **O_rigin**

_Analytically_, in an **even function**, f(−x) = \_\_\_\_\_

**f(x)**

_Graphically_, an **even function** will **reflect** about \_\_\_\_\_.

**the y-axis**

Another way to **describe** an **odd function** is \_\_\_\_\_.

**odd symmetric**

Another way to **describe** an **even function** is \_\_\_\_\_.

**even symmetric**

Any * *\_\_\_\_\_** with * *all even powers** is * *even symmetric**.

**polynomial**

Any * *polynomial** with * *all odd powers** is * *\_\_\_\_\_**.

**odd symmetric**

Any * *polynomial** with * *\_\_\_\_\_** is * *odd symmetric**.

**all odd powers**

_Analytically_, f(x) is **non-decreasing** over interval I = (a, b), if, for all **a \< x₁ \< x₂ \< b**, **\_\_\_\_\_**.

**f(x₂) _\>_ f(x₁)**

_Analytically_, f(x) is **non-decreasing** over interval I = (a, b), if, for all **a \< x₁ \< x₂ \< b**, **\_\_\_\_\_**.

**f(x₂) _\>_ f(x₁)**

_Analytically_, f(x) is **\_\_\_\_\_** over interval I = (a, b), if, for all **a \< x₁ \< x₂ \< b**, **f(x₂) _\>_ f(x₁)**.

**non-decreasing**

_Analytically_, f(x) is **\_\_\_\_\_** over interval I = (a, b), if, for all **a \< x₁ \< x₂ \< b**, **f(x₂) _\<_ f(x₁)**.

**non-increasing**

_Analytically_, f(x) is **\_\_\_\_\_** over interval I = (a, b), if, for all **a \< x₁ \< x₂ \< b**, **f(x₂) \> f(x₁)**.

**increasing**

_Analytically_, f(x) is **increasing** over interval I = (a, b), if, for all **\_\_\_\_\_**, **f(x₂) \> f(x₁)**.

**a \< x₁ \< x₂ \< b**

_Analytically_, f(x) is **increasing** over interval I = (a, b), if, for all **a \< x₁ \< x₂ \< b**, **\_\_\_\_\_**.

**f(x₂) \> f(x₁)**

_Analytically_, f(x) is **\_\_\_\_\_** over interval I = (a, b), if, for all **a \< x₁ \< x₂ \< b**, **f(x₂) \< f(x₁)**.

**decreasing**

_Analytically_, f(x) is **decreasing** over interval I = (a, b), if, for all **a \< x₁ \< x₂ \< b**, **\_\_\_\_\_**.

**f(x₂) \< f(x₁)**

Over **x ∈ [0, 1]**, this function is _[increasing / decreasing / neither increasing nor decreasing]_.

**increasing**

Over **x ∈ [1, 2]**, this function is _[increasing / decreasing / neither increasing nor decreasing]_.

**decreasing**

Over **x ∈ [0, 2]**, this function is _[increasing / decreasing / neither increasing nor decreasing]_.

**neither increasing nor decreasing**

_Pitfall_: When describing whether a **function** is **increasing or decreasing**, the pitfall is **\_\_\_\_\_**.

**the interval**

**y = mx + b** is a **\_\_\_\_\_** function.

**linear**

**y − y₀= m(x − x₀)** is a **\_\_\_\_\_** function.

**linear**

**f(x) = \_\_\_\_\_** is a **power** function.

**{ x^p | p** **∈ ℝ }**

**f(x) = { x^p | p** **∈ ℝ }** is a **\_\_\_\_\_** function.

**power**

Which are power functions? ``` f(x) = **x^−1/2** g(x) = **x^π** h(x) = **x^√(−1/2)** ```

Yes Yes No The exponent **must be a real number**, although it can be rational or irrational.

The **domain** of a **power function** is \_\_\_\_\_.

**D = ℝ**

The **domain** of **rational function** R(x) = _n(x)_ d(x) is \_\_\_\_\_.

**D = { x ∈ ℝ | d(x) ≠ 0}**

_Plainly_, how do you determine the **domain** of a **rational function**?

**Set the denominator equal to zero.** The domain is everything but that.

What is this asking about?

_Analytically_, a function f(x) has a **\_\_\_\_\_** at level y = L if **lim_x→∞ = L** or **lim_x→−∞ = L**

**horizontal asymptote**

_Analytically_, a function f(x) has a **horizontal asymptote** at level y = L if **\_\_\_\_\_** or **\_\_\_\_\_**.

**lim_x→∞ = L** or **lim_x→−∞ = L**

_Plainly_, what is **Step 1** to finding a **horizontal asymptote**?

Look to * *highest-degree term** in * *numerator and denominator**

_Plainly_, what is **Step 2** to finding a **horizontal asymptote**?

See if the highest-degree terms in numerator and denominator **approach a finite number.**

*"When you're taking a limit and the numerator and denominator are of the same degree, you're typically going to get a finite number."*