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Section 2.1

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.

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Section 2.1

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  1. Section 2.1

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

  3. Algebra 1 Review! Remember slope from Algebra 1?

  4. 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.

  5. 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 x1into 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 x1into the a function or acceleration function.

  6. 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:

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

  8. 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.

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

  10. Question 1: Average velocity is defined as the ratio of which two quantities?

  11. Question #2 Average velocity is equal to the slope of a secant line through two points on a graph. Which graph??

  12. Question #3 Can instantaneous velocity be defined as a ratio? If not, how is instantaneous velocity computed?

  13. Question #4 What is the graphical interpretation of instantaneous velocity at a moment t = t0??

  14. Question #5 What is the graphical interpretation of the following statement: The AROC approaches the IROC as the interval [x0, x1] shrinks to x0.

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