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=x2.
a=1 , so
m= limx→1 f(x) – f(1)/ x – 1 = limx→1 x2 – 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.