4 Applications of Differentiation

In this module, you will learn about the applications of differentiation.

Lessons

• • Lesson 4.1: Derivatives as Rates of Change

  • Lesson 4.2: Implicit Differentiation

• Lesson 4.3: Related Rates

  • Lesson 4.4: Linear Approximation and Differentials

Module Start & End Dates

Please check the syllabus for module start & end dates.

Show Your Work Problems

Module 4 assignment

Other Information

Project 3: Newton's Method

The linearization of a function at x=a (the tangent line to a function at x=a), a very simple idea, is the basis behind a powerful method for approximating the solutions to equations of the form f(x)=0, called Newton's Method.

Deriving Newton's Method: Suppose that r is a solution of f(x)=0 where f(x) is a differentiable function on some interval about r. Suppose that x0 is an initial estimate of r that might have been obtained by some preliminary analysis. To hopefully obtain a better approximation to r, the tangent line to the curve y=f(x) at x=x0 is drawn and the x-intercept of that tangent line is selected as the next estimate, x1.

To hopefully improve upon the approximation x₁, we repeat the process. We find the tangent line to the curve y=f(x) at x=x₁ and use the x-intercept of that tangent line as the next estimate, x₂.

Continuing in this fashion, we obtain a sequence of approximations x₁,x₂,x₃,... that ideally get closer and closer to the root r. In general, after n steps, the line tangent to the curve at x=x_n is used to find an updated approximate value, x_n+1.