Find the point on the curve y=cosx closest to the point (0,0)?

1 Answer
May 13, 2018

The point is (0,1)

Explanation:

By the distance formula, the distance between (0, 0) and the graph of y = cosx is

d= sqrt((x -0)^2 + (cosx - 0)^2)

d = sqrt(x^2 + cos^2x)

To find the minimum distance, we need to differentiate.

d' = (2x - 2cosxsinx)/(2sqrt(x^2 + cos^2x))

d' = (x - cosxsinx)/sqrt(x^2 + cos^2x)

We wish for this to be the smallest possible, thus we need d' = 0.

0 = (x - cosxsinx)/sqrt(x^2 + cos^2x)

0 = x - cosxsinx

Use a graphing application to solve and find that x = 0 is the only solution.

There derivative is negative when x < 0, and positive when x > 0, therefore, the minimum distance will occur when x = 0. The value of cosx at x = 0 is 1, therefore the closest point is (0, 1). Let's examine graphically:

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Hopefully this helps!