Question

Use linear approximation, i.e. the tangent line, to approximate 3√27.4as follows:

Let f(x)=3√x. The equation of the tangent line to f(x) at x=27 can be written in the form y=mx+b.
where m is:
and where b is:
Using this, we find our approximation for 3√27.4 is

Answer 1

To approximate 3√27.4 using linear approximation, find the equation of the tangent line to the function f(x) = 3√x at x = 27. Determine the slope, m, and the y-intercept, b, of the tangent line. Use the equation of the tangent line to approximate 3√27.4 by substituting x = 27.4 into the equation.

Step-by-step explanation:

To approximate the value of 3√27.4 using linear approximation, we need to find the equation of the tangent line to the function f(x) = 3√x at x = 27. We can find the slope, m, of the tangent line by taking the derivative of f(x) and evaluating it at x = 27. Then, we can find the y-intercept, b, by substituting the slope and the coordinates (27, f(27)) into the equation y = mx + b. Using the equation of the tangent line, we can approximate 3√27.4 by substituting x = 27.4 into the equation.

First, let's find the slope, m, of the tangent line. The derivative of f(x) = 3√x can be calculated using the power rule of derivatives:

f'(x) = (1/2) * 3 * x^(-1/2) = 3/(2√x)

Substituting x = 27 into the derivative, we get:

f'(27) = 3/(2√27)

Next, let's find the y-intercept, b, of the tangent line. We can use the point-slope form of a linear equation and substitute the slope and the coordinates (27, f(27)) into the equation:

f(27) = 3√27 = 3*3 = 9

So, we have the point (27, 9) on the tangent line. Substituting into the equation, we get:

9 = (3/(2√27))*27 + b

Simplifying,

9 = 27/(2√27) + b

Multiplying both sides by 2√27 to isolate b,

9 * 2√27 = 27 + b

18√27 - 27 = b

Now we have the equation of the tangent line as:

y = (3/(2√27))x + (18√27 - 27)

Finally, to approximate 3√27.4, we substitute x = 27.4 into the equation:

y ≈ (3/(2√27)) * 27.4 + (18√27 - 27) = 3√27.4