12.3 The Tangent Line Problem

Slope is the rate at which a line rises or falls, but how do we determine the slope for a curve?

Since curves have different rates of rising or falling, different points on the curve have different slopes. 

You can determine the slope of a point on a curve by creating a tangent to that point.

A tangent is a line that intersects at only one point.

You can find the tangent line by one of two ways: Approximation or Using the Derivative.

Finding the Tangent line by Approximation can be helpful but it is not accurate, it's just to get an estimate.

For Approximation you choose a point on the curve and estimate the slope by using the point and the point next to it to estimate the rate of change y/x to get the slope. then with the slope you use point slope form (Y-Y1=M(X-X1) then simplify into the standard Y=MX+B to get the formula of the tangent line.

The other, actually accurate way of finding the tangent line is by using the Derivative.

First you must find the difference quotient ((f(x+h)-f(x))/h) of the function of the curve 

from the difference quotient you can determine the Derivative(denoted as ) 

since the limit is of h going to 0, you substitute 0 in for every h in the simplified formula. 

Next, you plug in the x value for the selected point into the Derivative which will give you the slope at that point on the curve. 

Finally, use point slope form (Y-Y1=M(X-X1) then simplify into the standard Y=MX+B to get the formula of the tangent line.

For example: Find the tangent of the equation  at the point (2,4)

Find the difference quotient  

Simplify 

Simplify 

Find the Derivative by substituting in 0 for h for the limit 

substitute in the x value of the point into the Derivative to find the slope at the point (2,4)

 use point slope form then simplify into standard to find the equation of the tangent