+ Section 2.1. + Average velocity is just an algebra 1 slope between two points on the position function.

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Presentation transcript:

+ 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. Y 1 is a value found by plugging x 1 into an equation…but what equation? Therefore, y 1 is a little vague.

+ If we call our position function s(x) as opposed to y, then s(x 1 ) is the same as y 1. It is easier to see that s(x 1 ) is the y value that was obtained by plugging x 1 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(x 1 ) is the y value associated with plugging x 1 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 IntervalAverage Velocity [0.5, 0.6] [.5, 0.55] [.5, 0.51] [.5, 0.505] [.5, ] S(.5) = -4 S(.6) = S(.55)=-4.84 S(.51) = S(.505) = S(.5001) =

+ 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?

+ Question #4 What is the graphical interpretation of instantaneous velocity at a moment t = t 0? ?

+ Question #5 What is the graphical interpretation of the following statement: The AROC approaches the IROC as the interval [x 0, x 1 ] shrinks to x 0.