Using differentials, find the approximate value of √37.

To approximate the value of √37 using differentials, we can start by identifying a nearby perfect square. The closest perfect square to 37 is 36, which is equal to 6². This means we can use the function f(x) = √x and evaluate it at x = 36.

Step 1: Calculate the derivative

The derivative of f(x) = √x is given by:

f'(x) = 1/(2√x)

Now, substituting x = 36:

f'(36) = 1/(2√36) = 1/12

Step 2: Determine the change in x

Next, we find the change in x:

Δx = 37 - 36 = 1

Step 3: Apply the differential

Using the differential formula, we can approximate the change in f:

Δf ≈ f'(36) * Δx

Substituting the values we found:

Δf ≈ (1/12) * 1 = 1/12

Step 4: Calculate the approximate value

Now, we can find the approximate value of √37:

√37 ≈ f(36) + Δf = 6 + 1/12

Converting 1/12 to a decimal gives us approximately 0.0833.

Final Result

Thus, the approximate value of √37 is:

√37 ≈ 6.0833