Derivation Flashcards
What is the gradient function?
The formula for the gradient of the tangent at any point (x)
What are some notations of the gradient function?
How is first principal’s derived?
Splitting it up into two points, with a distance of h (later made to be 0) and then finding tangent gradient
What is the first principles formula for the gradient of the tangent line?
What does this mean?
The limiting value (output ) as h tends towards 0
What does the derivative tell you ?
How quickly the gradient function is increasing/ decreasing at that point
What is the quick way to differentiate when the base is x and the power is a constant ?
What is the derivative of a constant?
0
Differentiate the equation
how would you manipulate given expressions into a sum of x ^n terms?
- turn roots into powers (fractions)
- split up into fractions
- Expand out brackets
- if x is in the denominator, write it as a negative power in the numerator
- Beware of numbers in denominators (don’t take co efficents of x up into numerator, leave down as a coefficent fraction
- if there are multiple brackets, expand the brackets
How would you find the equation of a tangent to the curve at a point?
- Differentiate the equation to get the gradient function
- Put x value into this gradient function =m
- Using m , y ,x firm an equation
What is a normal to the curve ?
The perpendicular line of the tangent at that point
What is the relationship between the tangent of a curve and the normal?
What is an increasing function ?
A function where f ‘ (x) > 0
NOT when f ‘ (x) = 0 as
Describe the intervals for which x is increasing or decreasing
What is a decreasing function?
When f ‘ (x) < 0
How would you show that a function is increasing?
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
what is the stationary point/ turning points?
where f ‘ (x) =0
what are the three types of turning points?
- local minima
- local maxima
- saddle point (point of inflection)
not all points of infliction are turning points
what does “local” minima/ maxima mean?
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
what is a point of inflection?
- increasing before the point is the point, increasing after
- decreasing before, decreasing after
the direction of lines/ gradient is the same before and after
how would you determine from the gradient if there is a
- local maximum point
- local minimum point
- point of inflection
- gradient [-] —-> [+]
- gradient [+]—->[-]
- gradient [+]—->[+], [-]—-> [-]
what does the second derivative tell you?
how quickly the gradient is increasing or decreasing
what does an increasing gradient look like?
- going from negative to positive
- tangent becomes less steeper
postive second derivertive
what does a decreasing gradient look like?
- tangent becomes steeper
negative second derivative
what does it mean when
1. f “ (a) >0
2. f “ (a) <0
3. f “(a) =0
- local minimum
- local maximum
- USELESSS instead substitute previous and pre values of x
How would you sketch this gradient of this graph (y=x) on y =f’(x)
X axis becomes gradient, all TPS turn into roots
Sketch the gradient of this graph
How many degrees smaller will the gradient function be than the original function ?
1
Sketch the gradient function of this graph
In general if the original function f(x) has any horizontal asymptotes how will this be shown in the ketch of the gradient function
As an asymptote to the x axis
Sketch gradient function of the following graphs
Sketch the gradient function of the graph that has vertical asymptotes
Vertical asymptotes stay the same in gradient graph aswell
Sketch the gradient function of the graph that has vertical asymptotes
Vertical asymptotes stay the same in gradient graph aswell
What are optimisation problems?
Trying to maximise/ minimise a value of a variable we can control
Notation
Make sure to discard invalid solutions
