Calculus - Tangent Lines Revisted

Tangent Lines Revisited

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

Draw

Sketch the graph of $$f\left(x\right)=x^2-2$$ , and the tangent line $$L\left(x\right)$$ when $$x=2$$

Applications

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

Why Would We Need This?

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.

Multiple Choice

Use a single iteration of Newton's Method to find the approximate solution of $$f\left(x\right)=x^2-2$$ . Let $$a=1$$

• $$x=\frac{1}{2}$$

• $$x=\frac{3}{2}$$

• $$x=-\frac{1}{2}$$

• $$x=0$$

A Non-Algebraic Function

Visualize the Problem

Multiple Choice

What function describes the area of a rectangle inscribed under the sine curve?

• $$A\left(x\right)=x\cdot\ \sin\ x$$

• $$A\left(x\right)=\left(\pi\ -x\right)\cdot\ \sin\ x$$

• $$A\left(x\right)=\left(\pi\ -2x\right)\cdot\ \sin\ x$$

• $$A\left(x\right)=\pi\ \cdot\ \sin\ x$$

Moving Forward

Multiple Choice

What is the derivative of $$A\left(x\right)=\left(\pi-2x\right)\cdot\ \sin\ x$$ ?

• $$\frac{\text{d}A}{\text{d}x}=\left(\pi-2x\right)\cdot\ \cos x$$

• $$\frac{\text{d}A}{\text{d}x}=-2\sin\ x+\left(\pi-2x\right)\cdot\ \cos x$$

• $$\frac{\text{d}A}{\text{d}x}=-2\sin\ x$$

• $$\frac{\text{d}A}{\text{d}x}=0\$$

Maximize the Function

Maximize the Function

How Do We Continue?

Applying Newton's Method

Multiple Choice

What is the derivative of $$f\left(x\right)=2\tan\ x+2x-\pi\$$ ?

• $$f'\ \left(x\right)=2\sec^2x+2$$

• $$f'\ \left(x\right)=2\sec^2x-2$$

• $$f'\ \left(x\right)=\sec^2x+1$$

Applying Newton's Method

Applying Newton's Method

x must satisfy:

Wrapping Up