Use linear interpolation to estimate the desired quantity. A...

Use linear interpolation to estimate the desired quantity. A sensor measures the position $f(t)$ of a particle $t$ microseconds after a collision as given in the table. $$\begin{array}{|l|l|l|l|}\hline t & 5 & 10 & 15 \\\hline f(t) & 8 & 14 & 18 \\\hline\end{array}$$ Estimate the position of the particle at times (a) $t=8$ and (b) $t=12$

Use linear interpolation to estimate the desired quantity.
A sensor measures the position $f(t)$ of a particle $t$ microseconds after a collision as given in the table. $$\begin{array}{|l|l|l|l|}\hline t & 5 & 10 & 15 \\\hline f(t) & 8 & 14 & 18 \\\hline\end{array}$$ Estimate the position of the particle at times (a) $t=8$ and
(b) $t=12$

Key Concepts

Linear Interpolation

Linear interpolation is a method used to estimate an unknown value by assuming that the change between two known values is linear. This means that the value being estimated lies directly on the straight line connecting the two known points. It is a fundamental interpolation technique often used in numerical analysis and data estimation for its simplicity and effectiveness.

Rate of Change (Slope)

The rate of change between two data points, also known as the slope, represents how much the dependent variable changes per unit change in the independent variable. In the context of linear interpolation, the slope is calculated from the two known values and then used to determine the estimated value at a point between them.

Piecewise Linear Approximation

Piecewise linear approximation involves dividing the data into segments and applying linear interpolation on each segment where the function is approximated by a straight line. This concept is particularly useful when dealing with multiple intervals or segments of data, allowing each interval to be handled individually based on its own rate of change.

Transcript

00:01 Okay, so the two things that we're going to take first before we begin this problem are the first two set of points, which is going to be 5, 8, and 10, 14.

00:14 Okay, so with this, i'm going to find the slope of the line that can go through these two points.

00:20 So i'm going to calculate the difference of the y coordinates, which is 14 minus 8, over the difference of the x coordinates, which is 10 minus 5.

00:29 So if we calculate this, we get six -fits.

00:33 So with this, we're going to use point slope to create our linear approximation equation.

00:40 So you could use any of these two points as the x -y coordinates.

00:46 I'm going to use the second because eight is closer to 10 than five.

00:51 But it doesn't really mind.

00:52 You can use any point.

00:54 So i'm going to do y -minus 14 is equal to six -fifths.

01:00 X minus 10.

01:03 Then from here i'm going to solve for y by bringing 14 to the other side.

01:06 So i have y equals six fifths times x minus 10 plus 14.

01:13 So this is our l of x or l of t.

01:16 I'm going to use l of x because it's a little bit easier to see since we've used x.