Calculus - Tangent Lines Revisted

Tangent Lines

Recall that a line L(x) is tangent to the curve of a function f(x) if, and only if, L(x) passes through two points which are infinitely close together.

Tangent Lines

Because tangent lines approximate the behaviour of a function, we can find the approximate solutions to a function by using tangent lines

Approximating Solutions

A tangent line is a straight line which approximates the behaviour of a function near a point.

Linear Approximation

Finding Newton's Method

Finding Newton's Method

Not All Functions Are Algebraic!

Sometimes it is impossible to find a solution by applying Algebra.

So we need another way to deal with these kinds of problems. As long as the function is differentiable, we can try to apply Newton's Method to approximate the solution numerically.

A Non-Algebraic Function

Moving Forward

Maximize the Function

Maximize the Function

How Do We Continue?

Applying Newton's Method

Applying Newton's Method

Applying Newton's Method

x must satisfy:

Wrapping Up