Which of the following represents the area of the region bounded by the graph of the function #f(x)=sqrt{x}# , the x-axis, the y-axis and the tangent to the graph of #f(x)# at point #(4, 2)#..?

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Answer 1 · 2017-10-10T03:23:56

I got #A = 2/3#

We have to start by finding the equation of the tangent. We know the point, so the information we need to find is the slope.

#f(x) = sqrt(x)#

#f'(x) = 1/2x^(-1/2) = 1/(2sqrt(x))#

This means that the slope is #f'(4) = 1/(2sqrt(4)) = 1/4#

So the equation is

#y - 2 = 1/4(x - 4)#

#y - 2 = 1/4x - 1#

#y = 1/4x + 1#

Now we trace the graphs to see what the region looks like.

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The area is given by

#A = A_"upper graph" - A_"lower graph"#

To compute this, we use the following expression:

#A = int_0^4 1/4x + 1 - sqrt(x) dx#

#A = [1/8x^2 + x - 2/3x^(3/2)]_0^4#

#A = 1/8(4)^2 + 4 - 2/3 4^(3/2)#

#A = 2 + 4 - 16/3#

#A = 2/3#

Hopefully this helps!