f(x) = sqrt(x+1)

a=3
change of x =.8 to estimate f(3.8)-f(3)

use linear approximation

To use linear approximation, we need to find the equation of the tangent line to the function f(x) at the point x=a=3. This equation can then be used to estimate the change in f(x) for a small change in x.

First, we need to find the derivative of f(x) with respect to x. The derivative of f(x) = sqrt(x+1) can be found using the power rule. The power rule states that if f(x) = x^n, then the derivative of f(x) is f'(x) = nx^(n-1).

Applying the power rule, the derivative of f(x) = sqrt(x+1) is f'(x) = (1/2)(x+1)^(-1/2).

Now, we have the derivative f'(x) = (1/2)(x+1)^(-1/2). Let's evaluate this derivative at x=a=3 to find the slope of the tangent line at that point.

f'(3) = (1/2)(3+1)^(-1/2) = (1/2)(4)^(-1/2) = (1/2)(1/2) = 1/4.

So, the slope of the tangent line at the point x=a=3 is 1/4.

Now, we can use the point-slope form of a line to find the equation of the tangent line. The point-slope form states that if a line has slope m and passes through the point (x1, y1), its equation is given by y - y1 = m(x - x1).

Using the point (x1, y1) = (3, f(3)) in the equation, we have y - f(3) = (1/4)(x - 3).

To estimate the change in f(x) for a small change in x, we can use the equation of the tangent line. Let's substitute x = 3.8 and x = 3 into the equation and find the difference.

For x = 3.8:
y - f(3) = (1/4)(3.8 - 3) = (1/4)(0.8) = 0.2.

So, f(3.8) - f(3) ≈ 0.2.

Using linear approximation, we estimate that the change in f(x) from x = 3 to x = 3.8 is approximately 0.2.