Calculus! I think I'm thinking too hard into these problems!?

Question

Find the point on the line y=4x+7 that is closest to the origin.

Answer 1

A line which is perpendicular to this line and which goes through the origin will cross this line at the point which is closest to the origin.

y=4x+7 has gradient 4, so the perpendicular line has gradient -1/4 (so that their product is -1). This means the line through the origin is y=-1/4*x. Then to find the point solve these equations simultaneously and you will get y=7/17 and x=-28/17.

Answer 2

The origin is 0,0.

First, substitute x=0.
Then solve 0=4x+7.

That will give you the two end-points of the line segment that will be closest to the origin.

You should be able to find the closest point from there.

Answer 3

No calculus is needed. The shortest distance from the origin is the perpendicular distance of the origin from the line. Does that help?

Answer 4

i'm no longer too beneficial what you're attempting to assert, or what you're questioning approximately. What you wrote replaced right into slightly ambiguous/imprecise. nicely,....it incredibly is been proved that, in any axiomatic equipment, there exist statements or propositions which will no longer be able to be justified/proved. incredibly, there are mathematical statements whose certainty we can't ever understand. as an occasion, seek for the Continuum hypothesis. It says that, many times talking, there is no different fee of infinity it incredibly is precisely between that of the organic numbers and that of the real line. it incredibly is been proved that the Continuum hypothesis will no longer be able to be the two proved or disproved decrease than the huge-unfold ZFC equipment of excellent judgment. So, your assertion "the respond is often the two applicable or incorrect," isn't unavoidably authentic.

Answer 5

Just draw perpendicular from origin and see where it meets.