We are calculating the slope of the function f(x) = sqrt(x) at x = 4.

f(4) = 2
f(4+h) = sqrt(4 + h)

[f(4+h) - f(4)]/h =?

You have to pick a small h,and calculate it. Then pick an even smaller value and calculate it again. That's called taking the limit as h-> 0

If you pick h=0.1, you get a slope of
0.2484
If you pick h=0.01, you get a slope of 0.2498
Try an even lower value of h and you will see the trend to the limit.
The exact value of the slope, using the formula for the derivative at x=4, is (1/2)/(sqrt 4) = 1/4

To calculate the slope of the function f(x) = sqrt(x) at x = 4, we can use the difference quotient formula:

[f(4+h) - f(4)]/h

First, let's substitute the values into the formula:

[f(4+h) - f(4)]/h = [sqrt(4+h) - 2]/h

To find the slope, we need to evaluate this expression for different values of h and observe the trend as h approaches 0.

Let's start by choosing h = 0.1:

[f(4+0.1) - f(4)]/0.1 = [sqrt(4 + 0.1) - 2]/0.1 = (2.049 - 2)/0.1 = 0.049/0.1 = 0.49

For h = 0.01:

[f(4+0.01) - f(4)]/0.01 = [sqrt(4 + 0.01) - 2]/0.01 = (2.0049 - 2)/0.01 = 0.0049/0.01 = 0.49

As we can see, the value of the expression is approaching 0.49. This indicates a trend towards the slope as h gets smaller and closer to 0.

To find the exact value of the slope, we can take the derivative of the function f(x) = sqrt(x) with respect to x:

f'(x) = (1/2)(x^(-1/2)) = 1/(2√x)

At x = 4, the derivative is:

f'(4) = 1/(2√4) = 1/(2 * 2) = 1/4

Therefore, the exact value of the slope at x = 4 is 1/4.

By choosing smaller and smaller values of h and observing the trend, we can approximate this exact value of 1/4 as shown above. Taking the limit as h approaches 0 gives us the precise slope value.