Lecture #11: Tangent Line Approximation & Newton’s Method

MATH 145 – Calculus I

Torin Quinlivan
tquinlivan@knox.edu

Knox College

October 7, 2025

Local Linearization

Recall that: If \(f(x)\) is differentiable at \(x = a\), the local linearization (or tangent line approximation) to \(f(x)\) near \(x = a\) is the function \[ y = L(x) \approx f(a) + f'(a)(x - a) \]

If \(f(x)\) is differentiable at \(x = a\), you can approximate it by its tangent line.

Example: Linear Approximation

Example: Estimate \(\sqrt{36.1}\) using linear approximation.

Example: Estimate \(f(2.3)\) using \(x=2\) for \(f(x) = x^3 e^{-x^2}\), if we know that \(f'(x) = e^{-x^2} (3x^2 - 2x^4)\).

Example: Suppose that \(g(x)\) is a function with \(g(130) = 46\) and \(f'(130) = 2\). Estimate \(g(125.5)\).

Linear Approximation and Algebra

Question: Where does the linear approximation \(l(x) = f(a) + f'(a) (x - a)\) intersect the \(x\)-axis?

Can we make this approximation better by moving the line? Can we iterate this?

Linear Approximation and Algebra

This iterative algorithm is known as Newton’s Method:

Newton’s Method for approximating solutions to \(f(x) = 0\):

  1. Use the graph to find a good starting point \(x_0\) near to the solution \(x\) to \(f(x) = 0\).
  2. Find \(x_{n+1} = x_n - \dfrac{f(x_n)}{f'(x_n)}\).
  3. Repeat until the difference \(x_{n + 1} - x_n\) is .

Example: Newton’s Method

Example: Estimate the roots of \(f(x) = x^2 - 2\) using Newton’s method.

Example: Newton’s Method

Example: Estimate the roots of \(f(x) = -x^2 + 2x + 2\) using Newton’s method.

Example: Newton’s Method

Example: Estimate the roots of \(f(x) = x^3 - x^2 - 2x + 1\) using Newton’s method.