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)..?
1 Answer
Oct 10, 2017
I got
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
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.
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!