12. Differentiation Flashcards
Gradient function
Derivative - a function to find the gradient at any point
Gradient function symbol for the curve f(x)
f’(x)
Gradient function symbol for the curve y = …
dx
Differentiation by first principles
h
Simplify and then remove all values with a h as it is infinitesimally small
What to write for each step of a differentiation by first principles?
f’(x) = lim
h–> 0
If y = ax^n dy/dx =
an x^n-1
How to differentiate with multiple parts?
Differentiate each individually
(1/3x^2)
1/3(x^-2)
Normal to a curve at point A
Perpendicular to the tangent of the curve at point A
Increasing functions
m is always >= 0
[a,b]
Means that x is between a and b
How to show that a function is always increasing or decreasing
Complete the square and use that anything squared is positive to prove
How to find an interval where a function is increasing/decreasing
Find the derivative and solve, sketch to show whether m is >= or <= 0
Second/ second order derivative
Differentiate twice
The rate of change of the gradient function
d^2y
——– or f’‘(x)
dx^2
Stationary Points
Where the gradient f’(x) = 0
Local minimum/maximum points
Not the actual min/max but in the region where the turning points are they are
Point of inflection
The point at which the curve of the line changes direction
How to find the type of stationary point
Use the second derivative
>0: minimum point
<0: maximum point
=0: substitute a value just above and just below x into f’(x), if both of those have the same sign it’s a point of inflection
Max/min value of y = f(x) on the graph y = f’(x)
Root at that x
Point of inflection of y = f(x) on the graph y = f’(x)
Turning point at that x
Positive gradient of y = f(x) on the graph y = f’(x)
Above x-axis
Negative gradient of y = f(x) on the graph y = f’(x)
Below x-axis
Vertical asymptote of y = f(x) on the graph y = f’(x)
Vertical asymptote at the same point
Horizontal asymptote of y = f(x) on the graph y = f’(x)
Horizontal asymptote on the x-axis
What to make sure you do when modelling?
Write dy/dx in terms of the right variables
How to solve modelling questions
- make 2 variables from your information
- make 2 equations from these, one equalling a constraint (number) and one equalling the value you are minimising/maximising
- use the constraint to eliminate one variable by rearranging in terms of the other
- substitute that into the other equation
- differentiate for the min/max and check which
