Sketch the region in the first quadrant enclosed by y=8/x,y=4x and y=1/4x
Get a piece of graph paper, and sketch the two lines (4x and x/4), then the inverse curve 8/x
To sketch the region in the first quadrant enclosed by the given equations, we need to find the points of intersection between the curves.
First, let's find the points of intersection between the curves y = 8/x and y = 4x.
Setting these two equations equal to each other, we have:
8/x = 4x
Multiply both sides by x to eliminate the denominator:
8 = 4x^2
Divide both sides by 4:
2 = x^2
Taking the square root of both sides:
±√2 = x
Since we are looking for the region in the first quadrant, we choose x = √2 (positive square root).
Now, let's find the points of intersection between the curves y = 8/x and y = 1/(4x).
Setting these two equations equal to each other, we have:
8/x = 1/(4x)
Multiply both sides by x to eliminate the denominator:
8 = 1/4
This is not possible because 8 and 1/4 are not equal. Therefore, these two curves do not intersect.
Next, let's find the points of intersection between the curves y = 4x and y = 1/(4x).
Setting these two equations equal to each other, we have:
4x = 1/(4x)
Multiply both sides by 4x to eliminate the denominator:
16x^2 = 1
Divide both sides by 16:
x^2 = 1/16
Taking the square root of both sides:
±1/4 = x
Since we are looking for the region in the first quadrant, we choose x = 1/4 (positive value).
Now, we have found the x-coordinate of the intersection points: x = √2 and x = 1/4.
To find the y-coordinate, substitute these x-values into any of the given equations.
For x = √2, using y = 8/x:
y = 8/(√2) = 4√2
For x = 1/4, using y = 4x:
y = 4(1/4) = 1
So, the intersection points are (√2, 4√2) and (1/4, 1).
The sketch of the region in the first quadrant enclosed by the curves y = 8/x, y = 4x, and y = 1/(4x) would include the following:
- The curve y = 8/x passes through the point (√2, 4√2) and approaches the x-axis as x increases.
- The curve y = 4x starts at the origin (0,0) and passes through the point (1/4, 1).
- The curve y = 1/(4x) does not intersect with the other curves.
Please note that the sketch can be done by plotting these points and drawing the corresponding curves.