Solving equations numerically

[burgerandcheese]

Q: Solve the equation xe^x = 1

This is how my book did it and I don't understand:

Rearraging, x + ln(x) = 0

Now we use the sign-change rule for f(x) = x + ln (x)
Then f(0.5) = -0.193.. and f(0.6) = 0.089...
The root is in between 0.5 and 0.6. Since |f(0.5)| is about twice |f(0.6)| the root is probably 2/3 the distance from 0.5 to 0.6, which is about 0.57

How did they conclude that "the root is probably 2/3 the distance from 0.5 to 0.6"? How did they get that 2/3? I don't see it? Please guide me

 

[Dr.Peterson]

They are making a linear interpolation.

They found two points where y has opposite signs, (0.5, -0.193) and (0.6, 0.089). Sketch those points, and suppose the graph of f is approximately a straight line between them. Geometrically, you'll see a pair of similar triangles, one below the axis and one above. The former has a height of about 0.2, the latter about 0.1. So the ratio of the horizontal dimensions is about 2:1, making the intersection with the axis about 2/3 of the total distance.

Did the book show a picture? I would expect that; perhaps if you show us that picture, we can help you understand it.

 

[burgerandcheese]

They are making a linear interpolation.

They found two points where y has opposite signs, (0.5, -0.193) and (0.6, 0.089). Sketch those points, and suppose the graph of f is approximately a straight line between them. Geometrically, you'll see a pair of similar triangles, one below the axis and one above. The former has a height of about 0.2, the latter about 0.1. So the ratio of the horizontal dimensions is about 2:1, making the intersection with the axis about 2/3 of the total distance.

Did the book show a picture? I would expect that; perhaps if you show us that picture, we can help you understand it.

The book didn't show any picture, but after your explanation I understand now, thank you

 

[mathdad]

They are making a linear interpolation.

They found two points where y has opposite signs, (0.5, -0.193) and (0.6, 0.089). Sketch those points, and suppose the graph of f is approximately a straight line between them. Geometrically, you'll see a pair of similar triangles, one below the axis and one above. The former has a height of about 0.2, the latter about 0.1. So the ratio of the horizontal dimensions is about 2:1, making the intersection with the axis about 2/3 of the total distance.

Did the book show a picture? I would expect that; perhaps if you show us that picture, we can help you understand it.

What is linear interpolation in simple non-math words?

 

[Dr.Peterson]

It's exactly what I described: "Sketch those points, and suppose the graph of f is approximately a straight line between them."

"Interpolation" means "filling in a gap in data ..."; "linear" means "... with a straight line".

Observing that you are also asking questions about equations of lines, I can point out that, given two points on a graph, you can interpolate by writing the equation of the line through those two points, using the point-slope form; this equation can be taken as a linear interpolation of the graph in that region.

In this problem, they "approximated an approximation" by rounding the values and using a rough proportion rather than an exact equation of the line.

 

[mathdad]

It's exactly what I described: "Sketch those points, and suppose the graph of f is approximately a straight line between them."

"Interpolation" means "filling in a gap in data ..."; "linear" means "... with a straight line".

Observing that you are also asking questions about equations of lines, I can point out that, given two points on a graph, you can interpolate by writing the equation of the line through those two points, using the point-slope form; this equation can be taken as a linear interpolation of the graph in that region.

In this problem, they "approximated an approximation" by rounding the values and using a rough proportion rather than an exact equation of the line.

Interestingly stated.