Core 4 Flashcards
Differentiation from 1st Principles strategy
1) substitute function into formula f’(x) = lim h-0 f(x+h)-f(x)/ h
2) Expand and simplify the expression
3) Let h tend to 0 and write down the limit of expression f’(x)
4) Find the value of the gradient at the point (a,b) on the curve evaluating f’(a)
Solving problems with polynomials with rational powers
1) Use the laws of algebra to make expression sum of the terms of form ax^n
2) Apply f(x) = ac^n f’(x) = nax^n-1 to each term to find derivative
3) Substitute any numbers required and answer
Answer problem involving rate of change
1) Read and understand the context, identifying any function or relationship.
2) Differentiate to get a formula for the relevant rate of change.
3) Evaluate under the given conditions
4) Apply this to answer the initial question, being mindful of the context and units.
Work out where the tangent or normal meets the curve
1) Differentiate the function of the curve
2) Equate this to the gradient of the tangent to normal (remember m(t)= 1/m(n)
3) Rearrange and solve for x
4) Substitute x in the function and solve for y
Work out the area bound between a tangent, a normal and the x-axis/y-axis
1) Work out the equation of the tangent and, from it, the equation of the normal
2) Work out where each line crosses the required axis. Lines cut the x-axis when y=0, and the y-axis when x=0
3) Sketch the situation if required
4) Use A= 1/2 x base x height, where the base is the length between the intercepts between the intercepts on the x-axis or y-axis and the height is the y-coordinate or x-coordinate respectively.
To identify main features of a curve
1) Work out where it crosses the axes (x=0 and y=0)
2) Consider the behaviour if the curves as X tends to infinity and identify any asymptotes
3) Work out the coordinates of the Turing points (dy/dx = 0) and determine their nature.
4) Use the information you have found to sketch the function
To optimise a given situation
1) Express the dependent variable (say y) as a function of the independent variable (say x)
2) Differentiate y with respect to x
3) Let the derivative be 0 and find the value of x that optimises the value of y
4) Examine the nature of the turning points to decide if its maximum or minimum
5) put turning point in context of the question
To solve problems that require integration
1) Identify the variables and express the problem as a mathematical equation
2) integrate
3) Use initial conditions to work out the constant of integration
4) Substitute c into question and answer
Calculate area under the curve
1) Make a sketch of the function, if there isn’t one provided
2) Identify the area that as to be calculated
3) Write down the definite integral associated with the area
4) Evaluate the definite integral and remember tat the area is always positive
Maximum turning point
d^y/dx^2<0
minimum turning point
d^2y/dx^2>0
Increasing function
dy/dx >0
Decreasing function
dy/dx<0
What to substitute when wanting to find a value for c?
A value of y and x
cosine rule
cosA = b^2 + c^2 - a^2 / 2bc
