Use linear approximation, i.e. the tangent line, to approximate 8.4^(1/3) as follows:

Question

Use linear approximation, i.e. the tangent line, to approximate 8.4^(1/3) as follows:

Let f(x)= x^(1/3) . The equation of the tangent line to f(x) at x=8 can be written in the form y=mx+c where m=1/12 and c=4/3:
Using this, find our approximation for 8.4^(1/3).

Answer 1

let's try that again:

the way I recall linear approximation is
f(x) = f(x0) + f'(x0)(x - x0)

here x = 8.4, x0 = 8
and f'(x) = 1/(3x^(-2/3))

so f(8.4) =(appr.) f(8) + f'(8)(8.4-8)
=appr 2 + (1/12)(.4) = 2 + 1/30
=appr 2.03333

(by calculator 8.4^(1/3) = 2.033

Answer 2

To use linear approximation to approximate 8.4^(1/3), we will first find the equation of the tangent line to f(x) = x^(1/3) at x = 8.

The equation of a line in slope-intercept form, y = mx + c, gives us the value of y (the function value) at any x (input value). In this case, the tangent line to f(x) at x = 8 can be written as y = (1/12)x + 4/3, where m = 1/12 and c = 4/3.

Now, we can use this linear equation to approximate 8.4^(1/3).

Substitute x = 8.4 into the equation y = (1/12)x + 4/3:

y = (1/12)(8.4) + 4/3
= 0.7 + 4/3
= 0.7 + 1.33
≈ 2.03

Therefore, the approximation for 8.4^(1/3) using the tangent line is approximately 2.03.

Answer 3

To approximate 8.4^(1/3) using linear approximation, we can first find the equation of the tangent line to the function f(x) = x^(1/3) at x = 8.

1. Find the derivative of f(x) with respect to x:
f'(x) = (1/3) * x^(-2/3)

2. Evaluate f'(8) to find the slope of the tangent line:
f'(8) = (1/3) * 8^(-2/3) = (1/3) * (1/2) = 1/6

3. Determine the y-coordinate of the point on the function f(x) at x = 8:
f(8) = 8^(1/3) = 2

4. Use the point-slope form of a line (y - y1) = m(x - x1) with m = 1/6, x1 = 8, and y1 = 2 to find the equation of the tangent line:
y - 2 = (1/6)(x - 8)
y = (1/6)x + 4/3

Now, we can use this equation to approximate 8.4^(1/3):

5. Plug in x = 8.4 into the equation of the tangent line:
y = (1/6)(8.4) + 4/3
y = 14/5 + 4/3

6. Simplify the expression:
y ≈ 2.8 + 1.3333333333333333 ≈ 4.133333333333333

Therefore, the approximation for 8.4^(1/3) using linear approximation is approximately 4.133333333333333.

Answer 4

the way I recall linear approximation is

f(x) = f(x_{0) + f'(x_{0)(x - x_{0)

here x = 8.4, x_{0 = 8

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

so f(8.4) =(appr.) f(8) + f'(8)(8.4-8)

=appr 2 + (1/12)(.4) = 2 + 1/30

=appr 2.03333

(by calculator 8.4^(1/3) = 2.033}}}}