Differentiation Flashcards
How do you find the derivative?
Prepare
Multiply coefficient by power
Take one away from power
Rearrange to remove fractional powers and negative powers
What do you do when a fraction, with an x on the bottom, has more than one value on the top. (When wanting to differentiate)
Split the fraction and solve individually
How do you simplify a fractional power?
Put x into a square root
Put bottom number in tick
Put top number outside
How do you simplify a negative power?
Take it downstairs
How do you calculate the rate of change?
Prepare Differentiate If x=\_\_\_\_\_ Substitute Solve
What is a tangent?
A straight line at intersects a curve once
How do you find the equation of a tangent?
Find y coord by subbing x value into the equation Find gradient : - differentiate - let x= - subbing in - solving Sub into y-b=m(x-a)
What pieces of information do you need to calculated the equations of tangents?
Two bits of information
- A point and a gradient
What does it mean if dy/dx < 0?
Curve is decreasing
What does it mean if dy/dx = 0?
Curve is stationary
What does it mean if dy/dx > 0?
Curve is increasing
How do you find out whether a curve is increasing, decreasing, or is stationary?
Prepare equation Differentiate Let x = \_\_\_\_\_ Substitute Solve dy/dx <=> 0 therefore, curve is inc/dec when x = \_\_\_\_\_
If a question asks,, Determine whether the curve is increasing, decreasing, or stationary at x = 10,, how do you answer?
Inc and Dec curves (Prepare Differentiate Let x = 10 Solve dy/dx <=> 0 therefore curve is strictly \_\_\_\_\_\_\_\_)
- any number squared is greater than 0*
When do stationary points occur?
When dy/dx = 0
What are the four types of stationary points?
Max TP
Min TP
Rising Point of Inflection
Falling Point of Inflection
How do you find stationary points and work out their nature of the points?
SP's : - Prepare - Differentiate - 'SP's occur when dy/dx = 0 - Set equation = 0 - Factorise for x values Partners: - Sub x values into original equations to find y co ordinates Table: - Put x terms into nature table - State (x,y) is a min/max/rising/falling
How do you complete a nature table?
Two columns, three rows
Top row - arrow, x, arrow, x, arrow
Middle row - under x’s = 0, under arrows = sub value into dy/dx, write + or -
Bottom row - draw shape of graph
Can be asked to draw graphs
If asked to draw a graph, how would you work out the points of interest?
Find TP by differentiating and SP’s =0
‘Cuts x-axis when y=0’ - factorise
‘Cuts y-axis when x=0’
What are closed intervals?
The maximum and minimum values present in a section of a graph
How do you find the closed intervals?`
Find stationary points Make sure x values fit in the given range Substitute all (calculated and given) values into original equation - use table function on calculate State max and min values 'max value is\_\_\_\_\_\_ when x=\_\_\_\_\_. min value is \_\_\_\_\_\_ when x=\_\_\_\_\_\_'
How do you sketch the graphs of derivatives?
Put stationary points on x-axis
Break the curve into sections
If line is going up, put it on top of the x-axis
If line is going down, put it underneath the x-axis
What topic is optimisation a part of?
Differentiation
