Linear Interpolation Formula: Derivation, Solved Examples

Ans. x = 4; x1 = 2; x2 = 6; y1 = 4; y2 = 7 are the known values.

The formula for interpolation is as follows:

y=y1+(x-x1)(y2-y1)x2-x1

I.e. y=4+(4-2)(7-4)(6-2)

y=112

Ans. x=4; x1=3; x2=5; y1=5; y2=6 (given)

Using linear interpolation formula,

y=y1+(x-x1)(y2-y1)(x2-x1)

Putting the values,

y=5+(4-3)(6-5)(5-3)

y=5+1(1/2)

y=5+0.5

y=5.5

Ans. Given, x=8; x1=5; y1=3.5; y2=6

Using linear interpolation formula,

y=y1+(x-x1)(y2-y1)(x2-x1)

Putting the values,

y=3.5+(8-5)(6-3.5)(10-5)

y=3.5+3(2.5/5)

y=3.5+3(1/2)

y=5

Ans. Interpolation is essentially guessing data points that are inside the range of the data you've previously been given and are in the middle of your existing data points. Extrapolation is the process of making educated guesses about data points that are outside the range of your data collection.

Interpolation is the process of estimating a value between two known values in a series of values. Interpolation is a strategy that allows us to estimate the values within a gap in graphical data when the data is accessible on each side of the gap or at a few specific places within the gap.

Ans. In mathematical models, there are various forms of interpolations. Here are a few examples:

(i) Linear Interpolation (using linear polynomials to build more data points).

(ii) Polynomial Interpolation (which entails taking a given set of data from the lowest degree polynomial and allowing it to pass through the points in the data set for Interpolation data).

(iii) Bilinear Interpolation (Interpolation type using two distinct variables leading to the formation of a 2D rectilinear grid; Interpolation is done in two different directions, one after the other).

(iv) Spline Interpolation (when an interpolant is a special place-wise polynomial).

Ans. Interpolation is commonly employed in statistical models for commercial and mathematical research since it aids in the prediction of future likely points in data analysis. One can determine where the general consistent trend will lead the pricing (of a product or service), prospective yield (and company growth) or acquire insights into the stock market under a specific market dynamic using the collected sets. In the bond market and the financial sector, financial experts have regularly used this strategy to derive logical conclusions.

Ans. When extending a straight line on the graph to predict the growth in the market price of fuel such as petrol or diesel with respect to global market trends such as currency value of a country or increment or decrement in share market values, one gets a data set of points that points towards an increase in fuel price as share market value falls or a currency value declines.

In the physical world, the mass and volume of a given substance may be estimated using a similar plotting method called Interpolation to determine how the volume of a substance varies when the mass of the material increases or decreases.

Ans. We'll choose the two data points closest to this number for our interpolation calculation because we're looking for the tree's height after eight years. So, in the interpolation calculation, we'll use (5, 3.5) and (10, 6) and then replace x = 8 into the formula. We have the following:

y-y=y2-y1x2-x1(x-x1)

y-3.5=6-3.510-5(x-5)

y-3.5=0.5(x-5)

y-3.5=0.5x-2.5

y=0.5x+1

y=0.5(8)+1

y=4+1

y=5

Therefore, the tree was about five feet tall after 8 years.

Ans. Since we want the population in the year 2009, using the year closest to 2009 i.e. (2000,142000) and (2010,153000) and then substitute x=2009 in the formula.

y-y1=y2-y1x2-x1(x-x1)

y-142,000=153,000-142,0002010-2000(x-2000)

y-142,000=11,00010(x-2000)

y-142,000=1,100(x-2000)

y-142,000=1,100x-2,200,000

y=1,100x-2,058,000

y=1,100(2009)-2,058,000

y=151,900