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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1 Answer
Oct 10, 2017

I got A = 2/3

Explanation:

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!