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A Level Maths > Derivation > Flashcards

Derivation Flashcards

1
Q

What is the gradient function?

A

The formula for the gradient of the tangent at any point (x)

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2
Q

What are some notations of the gradient function?

A
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3
Q

How is first principal’s derived?

A

Splitting it up into two points, with a distance of h (later made to be 0) and then finding tangent gradient

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4
Q

What is the first principles formula for the gradient of the tangent line?

A
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5
Q

What does this mean?

A

The limiting value (output ) as h tends towards 0

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6
Q
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7
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8
Q
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9
Q

What does the derivative tell you ?

A

How quickly the gradient function is increasing/ decreasing at that point

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10
Q

What is the quick way to differentiate when the base is x and the power is a constant ?

A
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11
Q

What is the derivative of a constant?

A

0

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12
Q
A
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13
Q

Differentiate the equation

A
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14
Q

how would you manipulate given expressions into a sum of x ^n terms?

A
  1. turn roots into powers (fractions)
  2. split up into fractions
  3. Expand out brackets
  4. if x is in the denominator, write it as a negative power in the numerator
  5. Beware of numbers in denominators (don’t take co efficents of x up into numerator, leave down as a coefficent fraction
  6. if there are multiple brackets, expand the brackets
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15
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16
Q
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17
Q
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18
Q

How would you find the equation of a tangent to the curve at a point?

A
  1. Differentiate the equation to get the gradient function
  2. Put x value into this gradient function =m
  3. Using m , y ,x firm an equation
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19
Q
A
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20
Q

What is a normal to the curve ?

A

The perpendicular line of the tangent at that point

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21
Q

What is the relationship between the tangent of a curve and the normal?

A
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22
Q
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23
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24
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25
Q
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26
Q

What is an increasing function ?

A

A function where f ‘ (x) > 0
NOT when f ‘ (x) = 0 as

27
Q

Describe the intervals for which x is increasing or decreasing

A
28
Q

What is a decreasing function?

A

When f ‘ (x) < 0

29
Q

How would you show that a function is increasing?

A

By getting some value of x squared —> as this will always be positive despite the value of x
And an added value
This can be done by normal, or by completing the square

30
Q
A
31
Q
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32
Q
A
33
Q

what is the stationary point/ turning points?

A

where f ‘ (x) =0

34
Q

what are the three types of turning points?

A
  • local minima
  • local maxima
  • saddle point (point of inflection)

not all points of infliction are turning points

35
Q

what does “local” minima/ maxima mean?

A

it is the largest/ smallest value within the vicinity. “global” manima/ maxima mean highest/ smallest value in the entire function

cubics can never be global as they tend to infinity

36
Q

what is a point of inflection?

A
  • increasing before the point is the point, increasing after
  • decreasing before, decreasing after

the direction of lines/ gradient is the same before and after

37
Q
A
38
Q

how would you determine from the gradient if there is a

  • local maximum point
  • local minimum point
  • point of inflection
A
  1. gradient [-] —-> [+]
  2. gradient [+]—->[-]
  3. gradient [+]—->[+], [-]—-> [-]
39
Q
A
40
Q

what does the second derivative tell you?

A

how quickly the gradient is increasing or decreasing

41
Q

what does an increasing gradient look like?

A
  • going from negative to positive
  • tangent becomes less steeper

postive second derivertive

42
Q

what does a decreasing gradient look like?

A
  • tangent becomes steeper

negative second derivative

43
Q

what does it mean when
1. f “ (a) >0
2. f “ (a) <0
3. f “(a) =0

A
  1. local minimum
  2. local maximum
  3. USELESSS instead substitute previous and pre values of x
44
Q
A
45
Q

How would you sketch this gradient of this graph (y=x) on y =f’(x)

A

X axis becomes gradient, all TPS turn into roots

46
Q

Sketch the gradient of this graph

A
47
Q

How many degrees smaller will the gradient function be than the original function ?

A

1

48
Q

Sketch the gradient function of this graph

A
49
Q

In general if the original function f(x) has any horizontal asymptotes how will this be shown in the ketch of the gradient function

A

As an asymptote to the x axis

50
Q

Sketch gradient function of the following graphs

A
51
Q

Sketch the gradient function of the graph that has vertical asymptotes

A

Vertical asymptotes stay the same in gradient graph aswell

52
Q

Sketch the gradient function of the graph that has vertical asymptotes

A

Vertical asymptotes stay the same in gradient graph aswell

53
Q

What are optimisation problems?

A

Trying to maximise/ minimise a value of a variable we can control

54
Q

Notation

A
55
Q
A
56
Q
A
57
Q
A

Make sure to discard invalid solutions

58
Q
A
59
Q
A

A Level Maths (25 decks)

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