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Flashcards in Calculus - Ohio State iTunes U class Deck (34)
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1
Q

What does a function do?

A

A function assigns to each number in its domain another number.

w1,l1

2
Q

What is the domain (for this course)?

A

“Unless I say otherwise, the domain consists of all numbers for which the rule makes sense.” Meaning, the numbers I can plug in.

w1,l1

3
Q

For f(x)=1/x, what is the domain of f?

A

The domain of f is all real numbers except 0.

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4
Q

If f(2)=4, f(3)=9, f(4)=16, what is f(x) likely to be?

A

f(x)=x2

Functions can also be written as what the function is supposed to do using English words:

f(x)=the square of x

or

f(x) =x squared

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5
Q

What is piecewise notation?

A

Here’s an example of piecewise notation:

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6
Q

Is 0 even or odd?

A

Even

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7
Q

What is wrong with

  • B(x)= some rearrangement of the digits of x*
  • ?*
A

It is too ambiguous to be the definition of a function.

B(352) could equal 325, 235, 532.

A function should take an input value and unambiguously return a single output value. So this B(x) is not a function.

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8
Q

What is the identity function?

A

f(x) = x

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(could use g or another letter, g(x) = x)

9
Q

When are two functions the same?

A

Two functions are the same when they have the same output for the same input. They must have the same domain. They do not have to have the same rule (sequence of operations).

Example 1: f(x) = x2/x and g(x)=x

are not the same because they do not have the same domain. Zero is not in the domain of f(x).

Example 2: f(x) = (1 + x)2 and g(x) = x2 + 2x + 1

are the same function. They do not have same rule/sequence of operations but they produce the same output for every input and have the same domain.

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10
Q

What is the a constant function?

A

f(x) = c

where c is a constant

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11
Q

What is a polynomial?

A

Polynomials are sums variables and exponent expressions. Each piece of the polynomial, each part that is being added, is called a “term”. Polynomial terms have variables which are raised to whole-number exponents (or else the terms are just plain numbers); there are no square roots of variables, no fractional powers, and no variables in the denominator of any fractions.

Here are some examples:
6x –2 This is NOT a polynomial term… …because the variable has a negative exponent.
1/x2 This is NOT a polynomial term… …because the variable is in the denominator.
sqrt(x) This is NOT a polynomial term… …because the variable is inside a radical.
4x2 This IS a polynomial term… …because it obeys all the rules.

Definition from Purple Math

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12
Q

How is absolute value written?

A

|x|

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13
Q

How are two functions combined?

A

If f(x) = 2x+1 and g(x) = x2

then you can combine the functions g(f(x)), meaning take the output of f(x) and put it into g(x).

if x = 3, then f(x) = 7 and g(7) = 49

so g(f(x)) = 49

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14
Q
A
15
Q

Why can’t f(x) = √x be written as

f(x) = number that squares to x?

A

Because it is ambiguous. If x = 9 then the answer could be 3 or -3. But the answer to f(x)=√x is only 3. So to write it in English it would be f(x)= the nonnegative number that squares to x.

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16
Q

Why is this not true?

√x2 = x

What is the true statement?

A

It doesn’t hold for negative numbers. The square root of a number is a positive number.

So if x= - 4 you end up with 4 = -4 which isn’t true.

The true statement is

√x2 = |x|

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17
Q

What is the domain of √x?

A

[0, ∞ )

Meaning numbers from 0, including 0, to infinity, but not including infinity because infinity is not a number.

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18
Q

What is the domain of f(x)= √(2x+4) ?

A

We know the domain of a the square root function is ≥ 0 so

2x + 4 ≥ 0

2x ≥ -4

x ≥ -2

(we didn’t have to change the direction of the sign since we didn’t divide by a negative number)

So the domain of g = [-2, ∞)

(square bracket means include 2, parenthesis means don’t include ∞ because it’s not a number)

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19
Q

What is the domain of

T(x) = √(1-x) + √ (1+x)

?

A

We need both of the square roots to be ≥ 0 so

1-x ≥ 0 & 1+ x ≥ 0

1 ≥ x & x ≥ -1

x ≤ 1 & x ≥ -1

Dom T = [-1, 1]

(it is in the range between and including -1 and 1)

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20
Q

How do you read…and what does it mean?

A

The limit of f of x as x approaches a is equal to L

means f(x) can be made as close to L as desired by making x close enough (but not equal) to a.

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21
Q
A

It equals 1. The limit of any constant function is the constant. (The function here is just the constant 1, e.g. f(x) = 1)

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22
Q

What is the Squeeze Theorem?

If g(x)f(x)h(x) for x near a

and lim of g(x) as x approaches a =the limit of h(x) as x approaches a and = L

then what is the limit of f(x)?

A
23
Q

What are the abscissa and ordinate of the ordered pair (0,8)?

A

Abscissa = 0

Ordinate = 8

24
Q

An ordered pair has abscissa 2 and ordinate −6.

What is the ordered pair?

A

(2, -6)

25
Q

What is the position of an ordered pair with an abscissa of -9 and an ordinate of -4?

A
26
Q
A

Up

27
Q
A

Down

28
Q

What is

A

DNE - Does not exist.

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29
Q
A

It’s actually DNE. Lesson: don’t use just one number and move the decimal.

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30
Q

If f(x) = sin(pi/x) and f(0.01) = 0, why shouldn’t you use a calculator?

A

The calculator has to estimate pi. This is basically sin(100*pi) so the calculator cannot really give the correct answer but we can mentally work it out…

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31
Q

What is the limit of a product?

A

The limit of a product is the product of the limits, provided the limits exist.

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32
Q

What is the limit of a sum?

A

The limit of a sum is the sum of the limits, provided the limits exist.

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33
Q

What is the limit of a quotient and how does the rule differ from the sum and product rules?

A

The limit of a quotient is the quotient of the limits, provided the limits exist, with the extra caveat that the limit of the denominator (lower number) cannot be 0.

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34
Q

Since the denominator is 0, is this limit undefined?

A

Since we can factor the numerator we can find the limit. Note that the factored and divided function is not the same as the original function, since the original function is not defined at 1. However, near the value of 1 they are the same function.

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