Linear Interpolation (Edexcel International A Level Further Maths)
Revision Note
Author
Mark CurtisExpertise
Maths
Linear Interpolation
What is linear interpolation?
Linear interpolation draws a straight line between the points A and B with coordinates and , where is an interval containing a root to the equation
As and have different signs, A and B are on either side of the x-axis
The approximation to the root is where this straight line cuts the x-axis
This is at the point X with coordinates
Points A, X and B lie on the same straight line
It helps to use the gradient formula
The gradient between and is
How do I apply linear interpolation using gradients?
This is best explained with an example
If the root of lies in the interval and and , then:
A is the point
B is the point
X is the point
This is the x-intercept of the line AB
A, X and B lie on the same straight line
The gradient of AX equals the gradient of XB
The order matters (not BX)
Use the gradient formula to set these equal
giving
Solve the equation to find
expand and solve to get
This process can be repeated to get the next approximation
This gradient method does not require a sketch
What other methods can I use to do linear interpolation?
You can sketch the points A, B and X on a diagram and use similar triangles either side of the x-axis to find an equation in
This leads to the same equation as the gradients above
Or you can use coordinate geometry to find the equation of the line AB
Use
Then find its x-intercept
Substitute in and solve for
Exam Tip
If the calculations involve lengthy decimals, you can use and to stand for their numbers in your working!
Worked Example
The equation where has a root in the interval .
Use linear interpolation once to find an approximation to the root, giving your answer to 3 decimal places.
Find and
It is easier to write them as and for now
Let A, B and X be the points , and
Use the gradient formula to find the gradient of AX
gradient of AX is
Use the gradient formula to find the gradient of XB
gradient of XB is
Set the two gradients equal to each other
Simplify and cross multiply
Expand and collect the terms on the right-hand side
Factorise out the , then make the subject
Substitute in and from above
Give your final answer to 3 decimal places
to 3 decimal places
If you did the working without and , keep lots of decimal places
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