Local Linearity
Let’s consider the polynomial
$$ f(x) = x^3 - 2x + 1 $$ We want to find an interval around $ x=1 $ where the tangent line approximation is within 0.1 of the actual function value.
Find the tangent line:
First, we find the derivative of $ f(x) $ : $ f’(x) = 3x^2 - 2 $
At $ x=1 $ , the function value is $ f(1) = 1^3 - 2(1) + 1 = 0 $ , and the slope of the tangent line is $ f’(1) = 3(1)^2 - 2 = 1 $ .
The equation of the tangent line at $ x=1 $ is: $ y - 0 = 1(x - 1) $ $ y = x - 1 $
Determine the interval of accuracy:
We want to find the interval around $ x=1 $ where $ |f(x) - (x-1)| < 0.1 $ . This inequality can be rewritten as:
$$ -0.1 < x^3 - 2x + 1 - (x - 1) < 0.1 $$ $$ -0.1 < x^3 - 3x + 2 < 0.1 $$
We can solve this inequality numerically. Let’s analyze the inequality in two parts:
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$ x^3 - 3x + 2 > -0.1 $ : This is equivalent to $ x^3 - 3x + 2.1 > 0 $ . We can use numerical methods (like a graphing calculator or software like Desmos) to find the roots of $ x^3 - 3x + 2.1 = 0 $ . One root is approximately $ x \approx 0.86 $ .
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$ x^3 - 3x + 2 < 0.1 $ : This is equivalent to $ x^3 - 3x + 1.9 < 0 $ . Again, using numerical methods, we find a root approximately at $ x \approx 1.14 $ .
Therefore, the interval where the tangent line approximation is within 0.1 of the function is approximately $ [0.86, 1.14 $ .
- Visual Representation:
y = x^3 - 2x + 1
y = x - 1
y = x^3 - 2x + 1.1
y = x^3 - 2x + 0.9
The Desmos graph shows the function $ f(x) = x^3 - 2x + 1 $ (blue), its tangent line at $ x=1 $ (red), and the boundaries representing the 0.1 error margin (green and purple). You can visually confirm that the tangent line stays within the 0.1 tolerance band around $ x=1 $ within the approximate interval we calculated. The exact interval boundaries would require more precise numerical methods. The local linearity approximation using the tangent line at $ x=1 $ is accurate to within 0.1 for approximately $$ x \in [0.86, 1.14 $$ .This demonstrates that local linearity holds true in a small neighborhood around the point of tangency, but the size of this neighborhood depends on the function and the desired level of accuracy.