Use linear approximation, i.e. the tangent line, to approximate cubed root { 1.1 }. The equation of the tangent line to f(x) at x = 1 can be written in the form y = mx+b.
Show your work. Do not use a calculator.
y = x^(1/3)
dy/dx = (1/3) x^-(2/3)
at x = 1, y = 1 and dy/dx = (1/3)=m
y = 1 + (1/3) (x-1)
y = (1/3) x + 2/3
at x = 1.1
y = (1/3)(1.1) + 2/3
y = (1/3) 3.1
To approximate the cubed root of 1.1 using linear approximation, we will use the tangent line to the function f(x) = x^(1/3) at x = 1.
First, we need to find the equation of the tangent line to the function f(x) at x = 1. To do this, we will find the slope of the tangent line (m) and the y-intercept (b).
1. Find the slope (m):
The slope of the tangent line is equal to the derivative of the function evaluated at x = 1.
f'(x) = (1/3)x^(-2/3) (using the power rule for differentiation)
Evaluate f'(x) at x = 1:
f'(1) = (1/3)(1)^(-2/3) = 1/3
Therefore, the slope of the tangent line is m = 1/3.
2. Find the y-intercept (b):
The y-intercept is the value of f(x) at x = 1. Substituting x = 1 into the function f(x) = x^(1/3):
f(1) = 1^(1/3) = 1
Therefore, the y-intercept of the tangent line is b = 1.
Now that we have the slope (m = 1/3) and the y-intercept (b = 1), we can write the equation of the tangent line:
y = mx + b
y = (1/3)x + 1
This is the equation of the tangent line to f(x) at x = 1.
Using this linear approximation, we can approximate the cubed root of 1.1 by evaluating f(x) at x = 1.1:
f(1.1) ≈ (1/3)(1.1) + 1
≈ 0.36666 + 1
≈ 1.36666
Therefore, the approximation of the cubed root of 1.1 using linear approximation is approximately 1.36666.
To use linear approximation, we need to find the equation of the tangent line to the function f(x) = x^(1/3) at x = 1.
Step 1: Find the slope of the tangent line (m):
The slope of the tangent line is the derivative of the function at x = 1. Let's find the derivative of f(x) = x^(1/3).
Using the power rule, we can differentiate f(x) with respect to x:
f'(x) = (1/3) * x^(-2/3)
Now substitute x = 1 to find the slope at x = 1:
m = f'(1) = (1/3) * 1^(-2/3) = (1/3)
Step 2: Find the y-intercept of the tangent line (b):
To find b, we need to find the value of f(x) at x = 1 and use the slope-intercept form of the equation.
f(x) = x^(1/3)
f(1) = 1^(1/3) = 1
So, the y-intercept (b) is equal to f(1) :
b = f(1) = 1
Step 3: Write the equation of the tangent line:
We now have the slope (m) and the y-intercept (b), so we can write the equation of the tangent line in the form y = mx + b.
y = (1/3)x + 1
Step 4: Use the tangent line to approximate the value of f(1.1):
Now, we can use the equation of the tangent line to estimate f(1.1) by substituting x = 1.1 into the equation:
y = (1/3)(1.1) + 1
y = 0.3667 + 1
y ≈ 1.3667
Therefore, using linear approximation, the approximation for the cubed root of 1.1 is approximately 1.3667.