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Calculus 30 AP > Unit 3 - Differentiation: Definition and Basic Derivative Rules > Flashcards

Unit 3 - Differentiation: Definition and Basic Derivative Rules Flashcards

1
Q

Equation of a tangent line to a graph at a point

A

f’(c)(x-c)+f(c)

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2
Q

How do you find the normal?

A

Find the slope of the tangent, and take the negative reciprocal.

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3
Q

What is the normal when the tangent line has a slope of 0?

A

The normal is a vertical line, represented as x=c

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4
Q

What is the equation for the average rate of change?

A

average rate of change = (f(x)-f(c)) / (x-c) where x=/=c

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5
Q

What letter represents distance in Calculus?

A

s

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6
Q

What is average velocity?

A

Change in position with respect to time.

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7
Q

Difference between speed and velocity?

A

Speed is absolute. Velocity is signed (can be positive or negative)

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8
Q

What is instantaneous velocity?

A

The limit of average velocity as Delta t approaches 0.

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9
Q

How do you find the derivative when given a table of data?

A

Use the two closest independent values, and find the secant. (this approximates the tangent)

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10
Q

How do continuity and differentiability relate?

A

If a function is differentiable at c, it is continuous as c (f must be continuous at c for f to be differentiable at c. If a function is not continuous at c, it is not differentiable at c. If a function is continuous at c, it is not necessarily differentiable.

Differentiability implies continuity, but continuity does not imply differentiability.

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11
Q

What is the derivative of e^x

A

e^x

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12
Q

Simple power rule

A

x^n = nx^(n-1) Does not work for exponentials (n^x).

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13
Q

Sum/Difference rule.

A

f(x)=g(x)+-h(x). If g(x) and h(x) are differentiable, then f(x) is differentiable and f’(x)=g’(x)+-h’(x)

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14
Q

Notations for derivatives.

A

f’(x)
y’
dy/dx
(d/dx)y
(d/dx)f(x)
Df(x)

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15
Q

Derivative of a constant.

A

If f is a constant function f(x)=a, then f’(x)=0

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16
Q

What is the derivative of the natural logarithm?

A

y=ln x, y’=1/x

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17
Q

What is the product rule?

A

(uv)’=uv’+u’v

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18
Q

What is the quotient law?

A

(u/v)’=(vu’-uv’)/(v^2)

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19
Q

Higher order derivatives

A

y^(n)
f^(n) (x)
d^(n) y / dx^(n)
d / dx^(n)

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20
Q

derivative of sin(x)

A

cos(x) over all real numbers

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21
Q

derivative of cos(x)

A

-sin(x) over all real numbers

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22
Q

derivative of tan(x)

A

sec^2 (x) where x does not equal (2k+1)/(2) * Pi, k an intiger

23
Q

derivative of cot(x)

A

-csc^2 (x) where x does not equal nPi, n an intiger

24
Q

derivative of csc(x)

A

-csc(x) * cot(x), where x does not equal nPi, n an intiger

25
Q

derivative of sec(x)

A

sec(x) * tan(x), where x does not equal (2k+1)/(2) * Pi, k an intiger

26
Q

Useful note for remembering some derivatives of trigonometric functions?

A

If the trigonometric function starts with a c, its derivative has a negative sign

27
Q

How do you find the derivative using its definition

A

find the limit as h approaches 0 of (f(x+h)-f(x)) / h

28
Q

What does it mean of f’(a) (f prime of x at a) does not exist?

A

The tangent might not exist, might be a vertical line, or might not be unique

29
Q

What does limit as x approaches c of (f(x)-f(c)/(f-x) equal?

A

The tangent line to the graph of f at a c, provided the limit exists

30
Q

Average rate of change equation

A

(f(x)-f(c))/(x-c), x =/= c

31
Q

What is average rate of change?

A

behavior over an interval

32
Q

What is instantaneous rate of change?

A

behavior at a number (not over a range)

33
Q

Average velocity equation

A

Delta s/ Delta t = (f(t1))-f(t0)) / (t1-t0)

34
Q

Instantaneous velocity equation

A

v = limit as delta t approaches 0 of Delta s/ Delta t = limit as t0 approaches t0 of (f(t1))-f(t0)) / (t1-t0)

35
Q

How to use the definition for derivative of a function at a number?

A

f’(c) = limit as x approaches c of ((f(x)-f(c))/(x-c)

36
Q

How to use the definition for a derivative function f’ of a function f?

A

f’(x) = limit as h approaches 0 of ((f(x+h)-f(x))/h

37
Q

What does it mean when asked to use “the definition” to find the answer?

A

Longer form. No shortcuts.

38
Q

What are the 3 ways a function cannot be differentiable at a point on a continuous function?

A

The limit from the left and right exist but are not equal (corner). The one-sided limits are both infinite (+ or -) (vertical tangent). The one-sided limits are infinite but one is positive and the other is negative (cusp).

39
Q

How do you know where the horizontal tangent lines of a graph of f are?

A

Where f’ is 0.

40
Q

What are the common errors you make?

A

Incorrectly reducing exponents. Forgetting negatives/adding them where they don’t exist. Improperly formatting quotient rules. READ THE QUESTION AND KNOW WHAT IT WANTS. Remember that some limits of trigonometric equations involve squares.

41
Q

Memory trick for quotient rules of derivatives.

A

Low d-high minus high d-low, draw a line and square below

42
Q

What do you need to remember about solving for x of a trigonometric equation?

A

Remember the domain. If none is given, give the general solution.

43
Q

How would you format a falling object’s speed?

A

Falling at abs(velocity). Falling refers to the direction, so signs are not needed.

44
Q

Position… Velocity… what comes next?

A

Position… Velocity… then acceleration, jerk and snap.

45
Q

Form 1 and Form 2

A

todo

46
Q

What should you watch out for when a question asks for a limit?

A

Check if it is using the definition of a derivative

47
Q

For the current unit, do vertical tangents exist

A

No.

48
Q

What does m (subscript tangent) represent?

A

Slope of the tangent. Equal to form 1 of the definition of a derivative

49
Q

What do you need to remember about graphs?

A

Check the scale of the axi.

50
Q

When asked to match a function’s graph to its derivatives graph, what do you do?

A

Remember that you don’t know the equations. Describe the behaviour of the graphs over intervals and points.

51
Q

Finding limit of a piece wise function at a point

A

Use form 1 (x -> c) from the left and right sided limits. Check for continuity as well

52
Q

Do cusps have derivatives?

A

No. Derivatives cannot be infinite (for now at least)

53
Q

How to determine continuity using differentiability.

A

Wherever f is differentiable, f is continuous. Where f is not differentiable, you need more information to determine continuity.

54
Q

Difference between at a point and value.

A

Point are like (x,y). Values are x=… units

Calculus 30 AP (10 decks)

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