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Barrons Calculus > Barrons Calc > Flashcards

Barrons Calc Flashcards

1
Q

A function is a function if …

A

Every x value has a single y value. Therefore f(-1) can not equal 1 and 2, but f(-1) and f(1) can equal the same y value, since each x value still has a single y value.

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2
Q

In bracket notation, parenthesis represent which type of bounds and brackets represent which type of bond

A

Parenthesis represent exclusive bounds. So you would use parenthesis when you have negative or positive infinity in the bound, since negative or positive infinity can never be inclusive. Brackets are used for inclusive, therefore [3,10] means that 3 and 10 are included in the range.

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3
Q

Graph X2 and what is its domain and range.

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4
Q

Graph X3. Domain and Range?

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5
Q

Graph rad(x). Domain and Range?

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6
Q

Graph 1/x. Domain and Range?

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7
Q

Graph |x|. Domain and Range?

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8
Q

Graph ln(x). Domain and Range?

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9
Q

Graph ex. Domain and Range?

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10
Q

Graph [[X]]. Domain and Range?

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11
Q

Y semetric means that the function is

A

The function is an even function. Therefore f(x) = f(-x)

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12
Q

Origin Symetry means the function is

A

The function is odd. f(-x) = -f(x)

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13
Q

What type of symetry does this function have?

A

Wrong bitch. The function is shifted 2 spaces to the right, therefore f(x) no longer equals f(-x).

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14
Q

What does the “+ 4” do to the function (x + 1)2 + 4

A

The function is shifted up 4 spaces.

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15
Q

What does rad(-x) compared to rad(x)?

A
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16
Q

What does -x2 look like in comparison to x2

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17
Q

Graph Cosx

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18
Q

Graph Sinx

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19
Q

Graph Cscx

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20
Q

Graph secx

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21
Q

Graph cotx

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22
Q

Graph tanx

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23
Q

List all 3 pythagorean identities

A
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24
Q

Classift Sinx, Cosx, and Tanx as either odd or even

A

Sinx - oddCosx - evenTanx - odd

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25
Q

List the double angle formulas

A
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26
Q

List the power reducing formulas

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27
Q

List the sum and difference formulas

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28
Q

Equals

A

f(g(x))

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29
Q

A function is one-to-one

A

If the function has 1 unique y value for each x value. Therefore in x2 f(-1) and f(1) producing the same y value of 1 would not be considered uniqe, x2 is not 1 to 1

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30
Q

In order to find the inverse of a function. The function must be

A

One to one. Think about it, a function is only a function if a x value yields only one y value. So if you were finding an inverse function (meaning you plug in y’s and x’s come out), you would need the function to be one to one, so that when you have your inverse function and plug in a y value, you do not get more than one x value.

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31
Q

What geometric test must functions pass?

A

Vertical Line Test

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32
Q

What geometric test must one to one functions pass

A

Horizontal Test

33
Q

Synthetic Division

A
34
Q

Formula for factoring cubics

A
35
Q

How would you represent Sec-1(x) in terms of cosine

A

Cos-1(1/x). Not 1/Cos-1(x)

36
Q

Cot-1(x) in terms of tan

A

Cot-1(x) = pi/2 -tan-1(x)

37
Q

lnaxm is equivalent to

A

m x lnax

38
Q

Simplify ln (3 * 5)

A

ln(3) + ln(5)

39
Q

Product Rule

A

(First function times the derivative of the second function) + (the second function times the derivative of the first function)

40
Q

Quotient Rule

A

[(2nd function times derivative of the 1st function) - (1st​ function times the derivative of the 2nd function)] / (2nd)2

41
Q

Derivative of efunction

A

efunction times derivative of the function

42
Q

Derivative of Cosx

A

-sinx

43
Q

Derivative of tanx

A

sec2x = (sec(x))2

44
Q

Derivative of Cotx

A

-csc2x

45
Q

Derivative of constantfunction (e.g. 5x^3)

A

constantfunction times ln(constant) times derivative of function. 5x^3 times ln(5) times 3x2

46
Q

Derivative of sin-1x

A
47
Q

Derivative of cos-1x

A
48
Q

Derivative of tan-1x

A
49
Q

Derivative of cot-1x

A
50
Q

Derivative of csc-1x

A
51
Q

Derivative of sec-1x

A
52
Q

The formula that defines a derivative

A
53
Q

A derivative of a vertical tangent at point c

A

Does not exist

54
Q

A derivative of the corner point at point c

A
55
Q

A derivative of a cusp at point c

A
56
Q

Which formula would you use to solve a problem like this “If f(x) = 5x and 51.002​ = 5.016, which is closest to f’(1)

A

The formula for a derivative. h = 0.002

57
Q

Implicit differentiation occurs when? What do you do in order to solve derivatives that require implicit differentiation?

A

Implicit differentiation occurs when a function is defined with both x and y as part of the function (e.g. x2 + cos(y)) instead of just y = x2. In order to solve differentiation you have to derivat the x normally and tag on a dy/dx every time you derive the y. Then solve for the dy/dx. If you are asked for the second derivative and given the function. First find the first derivative. Then substitute the value you got for dy/dx from the first derivative into the second derivative. You can have y in your answer for an implicity derived function.

58
Q

How does the derivative of a function relation to the derivative of the function’s inverse? What formula?

A

The derivative of an inverse function is the reciprocal of the derivative of the function, since the derivative of an inverse function is represented by dx/dy not dy/dx. (f-1)’ (x) = 1/ f’(f-1(x))

59
Q

Where is the tangent to the curve 4x2 + 9y2 = 36 vertical?

A

* The tangent line is found by the derivative, so this problem involves deriving in some way * So go ahead and derive the function (you will need to use implicit differentiation) * Once you have a value for dy/dx, you can move on to finding when the tangent line is vertical. * A vertical line is given by f(y) = some constant, because regardless of the y value you plug in, the x value will always be 0. * Therefore take the inverse of the dy/dx, which is dx/dy, by finding the reciprocal of dy/dx. Find when dx/dy = 0, then set this y value equal to 4x2 + 9y2 - 36

60
Q

What is the Mean Value theorem?

A

If the function is continuous between [a,b]. Then there must be some point (lets call this point x), where x’s derivative is equal to the average slope of a line connection points (a, y) and (b, y). This information is useful if you those problems that relate to getting a ticket for speeding. If you are travelling down a road and at point A, the police measure your speed to be 60 and then at point B, the police measure your speed to be 65, the time that it took you to travel from point A to point B (5 miles) is .25 hours. Then they can guarantee that at one point along your path from point A to B, you were going (5)/.25 mph at some point. In order to find out whether the Mean Value Theorem is satisfied check whether the function is continuous and then derive the function and check whether the are any values of [a, b] that make the derivative divided by 0 or any other discontinuity

61
Q

Define Rolle’s Theorem

A

Rolle’s Theorem is a special case of the the mean value theorem. The roles theorem claims that if a function is continuous between [a,b] and f(a) = f(b), then at some point (lets call this point x) f’(x) = 0, since the graph has to change direct (slope change from positive to negative or negative to positive) in order to return to (b, f(b)).

62
Q

Exponential Functions

A

a^0 = 1a^1 = aa^m * a^n = a^m+na^m / a^n = a^m-n

63
Q

In order for there to be a derivative at point C, then

A

* The function must be continuous at that point. Therefore no skips, jumps, holes.

64
Q

What is this question asking? “Does limit as x approaches 1 exist in f(x)?”

A

The question is asking if from the left and right side of the function approach the same value. This same value does not have to be defined.

65
Q

Chapter 3, practice question 50.lim x2 times sin(1/x)as x approaches infinity

A

You can not just do sin(1/

66
Q

What are considered critical values for the 1st derivative

A

when the 1st derivative = 0 or where the 1st derivative does not exist

67
Q

Point-slope formula of slope

A

y - y1 = f’(x1)(x - x1)

68
Q

What is “find the line normal to the curve a point P” asking?

A

The normal line means the line that is perpendicular to the tangent at point P. Find the f’(x1) and before plugging it into the point slope form, plug in the inverse so 1/f’(x1).

69
Q

Vertical Tangent Line means that the derivative

A

the derivative does not exist

70
Q

Local vs Global Extrema

A

There can be multiple local extreme at which the first derivative = 0, but only 1 global extrema at which the first derivative = 0

71
Q

What is the global min for |x|

A

x = 0, No need to derive, just look at the graph of |x|

72
Q

Procedure for sketching a curve

A
  1. Find the x intercepts
  2. Figure out whether the curve has x,y, or origin symmetry
  3. Find vertical and horizontal asymptotes
  4. End behavior
  5. Points of discontinuity
  6. Extrema
  7. Concavity
73
Q

Where to look for extrema

A

The beginning of the interval, the end of the interval, and where the at which the first derivative = 0

74
Q

Draw Unit Circle

A
75
Q

Difference quotient for finding specific point

A
76
Q

If a function is continuous, this means what about the derivatives at each point of the function?

A

A function being continuous does not guarantee that the function has a derivative at each point, because the function could have a cusp at some point, and corner points are not differentialable.

77
Q

If a function is differentiable at every point, this means what about the continuity of the function?

A

A differentiable function guarantees that the function is continuous

78
Q

Why are points at which vetical tangent lines exist undifferentiable?

A

Because vertical tangent lines do not have slopes that exist. Rise over run. The run is 0, so you would be dividing by 0 to find the slope of a vertical tangent line.

Barrons Calculus (2 decks)

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