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!