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Answer to Question #4905 in Calculus for klinc

Question #4905
find the point on the curve y= cosx closest to the point (0,0)
Expert's answer
Distance formula:
d = SQRT( (x1-x2)^2 + (y1-y2)^2 )
Here, x1 = y1 = 0.
d = SQRT( x^2 + y^2 )
y = cos(x) - equation of our curve
d = SQRT( x^2 + (cos(x))^2 )
f(x) = x^2 + (cos(x))^2

f'(x) = 2x - 2cos(x)sin(x) = 2x - sin(2x)
Set f'(x) to zero:
2x - sin(2x) = 0
2x = sin(2x)
2x = 0, or x=0 - the unique solution that minimize the distance
cos(0) = 1

So, the point on the curve y=cosx closest to the point (0,0) is point (0,1).

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Comments

Assignment Expert
16.01.14, 17:22

Take a look: the distance d = SQRT(x^2 + (cos(x))^2) and its square d^2=x^2 + (cos(x))^2 reaches the minimal value simultaneously. Thus one can find minimum of d^2 because it looks easier.

caroline werner
15.01.14, 05:26

how did you get rid of the SQRT after the 5th step? you can't just drop that trash....

Assignment Expert
05.04.13, 14:24

2x = sin(2x)
Applying double angle formula we get
2x = 2sinx cosx
or
x=sin x cos x
This is a transcendental equation which can be solved graphically. This equation has unique solution x=0.

imran sabir
04.04.13, 16:24

how did you go from

2x = sin(2x)

to

2x = 0.

How six(2x) =0?

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