Use the differential to approximate the expression. Then use a calculator to approximate the​ quantity, and give the absolute value of the difference in the two results to four decimal places.

Question

Sqrt61

Use the differential to approximate the expression. Then use a calculator to approximate the​ quantity, and give the absolute value of the difference in the two results to four decimal places.

Answer 1

Hi Erica,

When using differentials to approximate a value, we draw a tangent line at a known point nearby and find the value along the tangent line instead of on the function itself. We don't know what sqrt(61), but a nearby value that we do know is sqrt(64).

Let's set f(x) = sqrt(x)

We know that f(64) = 8

We can take the derivative of this function to find the slope of the tangent line:

f'(x) = 1/2(x)^(-1/2)

Since we want to draw the tangent line at x = 64, plug in 64 for x.

f'(64) = 1/2*(64)^(-1/2)

= 1/2 * 1/8 = 1/16

So the slope of our tangent line is 1/16, and we know that (64,8) is a point on that line. Using Point-Slope Form, we can determine the equation of the line.

y - 8 = (1/16)*(x - 64)

y = 1/16*x + 4

We're interested in approximating sqrt(61), so plug 61 in for x on your tangent line equation.

y = 1/16*61 + 4

= 125/16 = 7.8125

The actual value of sqrt(61) according to your calculator is 7.8102, so the difference is 0.0023.

We naturally over-approximated the true value a bit, since the tangent line sits a little bit above the curve of the function.

Hope this helps!