### Section 2.7 Tangents and Velocities

This section deals with tangents and velocities, which are rates of change. Tangent lines are lines that touch a point on a curve only once. There are two formulas that can be used to calculate the tangent. If a curve has the equation

*f*(

*x*), in order to find the tangent line to the curve at a point P (

*a, f*(

*a*)), then consider a nearby point Q (

*x*,

*f*(

*x*)), and compute the slope of the secant line PQ(a secant line is a line that passes through both points) by using:

*mPQ =*limx→a

*f*(

*x*) –

*f*(

*a*) /

*x*–

*a*

or if

*h*=

*x*–

*a*then

*m*= limh→0

*f*(

*a + h*) –

*f*(

*a*) /

*h*

This is also known as the

*derivative*. The terms slope of a tangent line, derivative, and instantaneous rate of change can be used interchangeably.

Here is an example of a tangent line:

Find an equation of the tangent line to the parabola

*y=x*2.

*a=*1 , so

*m=*limx→1

*f*(

*x*) –

*f*(1)/

*x*– 1 = limx→1

*x*2 – 1/

*x*– 1

= limx→1 (

*x*+ 1) (

*x*– 1) /

*x*– 1

= limx→1 (

*x*+ 1) = 1 + 1 = 2

Using the point slope form to find the equation of the line:

*y*– 1 = 2(

*x*– 1)

The average velocity is displacement over time.

*f*(

*a + h*) –

*f*(

*a*) /

*h*Do not confuse this with the derivative.

Here is an example:

Suppose a car drives 10m in 5 sec and 15m in 10 sec. Find the average velocity between the two points.

Take the displacement and put it over the time: 15-10/(10-5) = 5/5 = 1 m / sec.

Here is a link to a useful site:

http://www.sosmath.com/calculus/diff/der01/der01.html

Reminder to Luke for the next posting.

## 1 Comments:

The link was very helpful

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