+ section 2.1. + average velocity is just an algebra 1 slope between two points on the position...

+

Section 2.1

+Section 2.1

Average velocity is just an algebra 1 slope between two points on the position function.

+Algebra 1 Review!

Remember slope from Algebra 1?

+Function Notation

So now that we are in Calculus, we can make our slope formula (average rate of change formula) more specific.

Y1 is a value found by plugging x1 into an equation…but what equation? Therefore, y1 is a little vague.

+If we call our position function s(x) as opposed to y, then s(x1) is the same as y1.

It is easier to see that s(x1) is the y value that was obtained by plugging x1 into the s function.

We use many different types of functions in calculus so we need to keep them straight and the easiest way to do this is to use function notation.

Examples: a(x1) is the y value associated with plugging x1 into the a function or acceleration function.

+

Example 1: A stone is released from a state of rest and falls to earth. Estimate the instantaneous velocity at t = .5 s by calculating average velocity over several small time intervals.

Position function is:

+

Time Interval

Average Velocity

[0.5, 0.6]

[.5, 0.55]

[.5, 0.51]

[.5, 0.505]

[.5, 0.5001]

S(.5) = -4S(.6) = -5.76S(.55)=-4.84S(.51) = -4.1616S(.505) = -4.0804S(.5001) = -4.0016

+Every time we find an Average Rate of Change, (AROC), we are finding the slope of a secant line… a slope between two points.

The closer the x-values get to one another, the closer we get to an Instantaneous Rate of Change (IROC), or the slope of the tangent line.

+Tangent Lines

The more secant lines you draw, the closer you are getting to a tangent line.

+Question 1:

Average velocity is defined as the ratio of which two quantities?

+Question #2

Average velocity is equal to the slope of a secant line through two points on a graph. Which graph??

+

Question #3

Can instantaneous velocity be defined as a ratio? If not, how is instantaneous velocity computed?

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Question #4

What is the graphical interpretation of instantaneous velocity at a moment t = t0??

+Question #5

What is the graphical interpretation of the following statement: The AROC approaches the IROC as the interval [x0 , x1] shrinks to x0.