Derivation

Question 1

What is the gradient function?

Answer 1

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

Question 2

What are some notations of the gradient function?

Answer 2

Question 3

How is first principal’s derived?

Answer 3

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

Question 4

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

Answer 4

Question 5

What does this mean?

Answer 5

The limiting value (output ) as h tends towards 0

Question 6

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Question 7

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Question 8

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Question 9

What does the derivative tell you ?

Answer 9

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

Question 10

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

Answer 10

Question 11

What is the derivative of a constant?

Answer 11

0

Question 12

Answer 12

Question 13

Differentiate the equation

Answer 13

Question 14

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

Answer 14

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

Question 15

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Question 16

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Question 17

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Question 18

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

Answer 18

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

Question 19

Answer 19

Question 20

What is a normal to the curve ?

Answer 20

The perpendicular line of the tangent at that point

Question 21

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

Answer 21

Question 22

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Question 23

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Question 24

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Question 25

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Question 26

What is an increasing function ?

Answer 26

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

Question 27

Describe the intervals for which x is increasing or decreasing

Answer 27

Question 28

What is a decreasing function?

Answer 28

When f ‘ (x) < 0

Question 29

How would you show that a function is increasing?

Answer 29

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

Question 30

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Question 31

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Question 32

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Question 33

what is the stationary point/ turning points?

Answer 33

where f ' (x) =0

Question 34

what are the three types of turning points?

Answer 34

* local minima * local maxima * saddle point (point of inflection) | not all points of infliction are turning points

Question 35

what does "**local**" minima/ maxima mean?

Answer 35

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

Question 36

what is a point of inflection?

Answer 36

* increasing before the point is the point, increasing after * decreasing before, decreasing after | the direction of lines/ gradient is the same before and after

Question 37

Answer 37

Question 38

how would you determine from the gradient if there is a * local maximum point * local minimum point * point of inflection

Answer 38

1. gradient [-] ----> [+] 2. gradient [+]---->[-] 3. gradient [+]---->[+], [-]----> [-]

Question 39

Answer 39

Question 40

what does the second derivative tell you?

Answer 40

how quickly the gradient is increasing or decreasing

Question 41

what does an increasing gradient look like?

Answer 41

* going from negative to positive * tangent becomes less steeper | postive second derivertive

Question 42

what does a decreasing gradient look like?

Answer 42

* tangent becomes steeper | negative second derivative

Question 43

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

Answer 43

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

Question 44

Answer 44

Question 45

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

Answer 45

X axis becomes gradient, all TPS turn into roots

Question 46

Sketch the gradient of this graph

Answer 46

Question 47

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

Answer 47

1

Question 48

Sketch the gradient function of this graph

Answer 48

Question 49

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

Answer 49

As an asymptote to the x axis

Question 50

Sketch gradient function of the following graphs

Answer 50

Question 51

Sketch the gradient function of the graph that has vertical asymptotes

Answer 51

Vertical asymptotes stay the same in gradient graph aswell

Question 52

Sketch the gradient function of the graph that has vertical asymptotes

Answer 52

Vertical asymptotes stay the same in gradient graph aswell

Question 53

What are optimisation problems?

Answer 53

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

Question 54

Notation

Answer 54

Question 55

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Question 56

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Question 57

Answer 57

Make sure to discard invalid solutions

Question 58

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Question 59

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