Determine the linear approximation of 21/2. Express your answer as a reduced fraction in the form a/b.
To determine the linear approximation of √2, we can use the concept of a tangent line approximation. The tangent line at a specific point on a curve can be used to approximate the value of the function near that point.
First, let's choose a point near √2 on the curve of the function y = √x. To simplify the calculation, we can use x = 4, which is close enough to 2. The square root of 4 is 2.
Now, let's find the equation of the tangent line at x = 4. To do this, we need the slope of the tangent line and the coordinates of the point where the tangent line touches the curve.
The slope of the tangent line at x = 4 can be found using the derivative of the function y = √x:
dy/dx = 1/(2√x)
When x = 4:
dy/dx = 1/(2√4) = 1/4
So, the slope of the tangent line is 1/4.
We also need the coordinates of the point where the tangent line touches the curve. At x = 4, the corresponding y value is √4 = 2.
Now, we can write the equation of the tangent line:
y - 2 = (1/4)(x - 4)
Simplifying this equation:
y - 2 = (1/4)x - 1
y = (1/4)x + 1
This equation represents the tangent line approximation of the curve y = √x near x = 4.
Now, let's use this tangent line to approximate the value of √2.
Substitute x = 2 into the equation of the tangent line:
y = (1/4)(2) + 1
y = 1/2 + 1
y = 3/2
So, the linear approximation of √2 is 3/2.