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Warmup

Warmup. Find k such that the line is tangent to the graph of the function. Warmup:. 3.1 Derivatives. Stock Market/ Economic Crash. Nope, not that kind of derivative. We write:. is called the derivative of at. “The derivative of f with respect to x is …”.

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Warmup

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  1. Warmup Find k such that the line is tangent to the graph of the function

  2. Warmup:

  3. 3.1 Derivatives Stock Market/ Economic Crash Nope, not that kind of derivative

  4. We write: is called the derivative of at . “The derivative of f with respect to x is …”

  5. There are many ways to write the derivative of Alternate Form of Derivative provided the limit exists

  6. “the derivative of f with respect to x” “f prime x” or “y prime” “the derivative of y with respect to x” “the derivative of f with respect to x” “the derivative of f of x”

  7. Note: dx does not mean d times x ! dy does not mean d times y !

  8. does not mean ! Note:

  9. does not mean times ! Note:

  10. The derivative is the slope of the original function. The derivative is defined at the end points of a function on a closed interval.

  11. A function is differentiable if it has a derivative everywhere in its domain. It must be continuous and smooth. Functions on closed intervals must have one-sided derivatives defined at the end points. p

  12. Graphing a Derivative on TI 83+ The proper notation for graphing the derivative is nDeriv(function,X,X).

  13. Graph the derivative of the function

  14. Graph of f(x) Make a table of approximations of slopes of tangent lines at the pts 6 1 0 -1.5 -1 0 1 Now lets take these values and make a graph of all the slopes ( f ‘ (x) graph )

  15. A’ ?’ B’ F’ C’ E’ D’ Connect them for your derivative graph

  16. Sketch

  17. Conceptual questions: Let y = g(x) be a function that measures the water depth in a pool x minutes after the pool begins to fill. Then g’(25) represents: • The rate at which the depth is increasing 25 minutes • after the pool starts to fill • II. The average rate at which the depth changes over • the first 25 minutes • III. The slope of the graph of g at the point where x = 25 • I only B) II only C) III only D) I and II • E) I and III F) I, II, and III

  18. The function y = f(x) measures the fish population in Lincoln Pond at time x, where x is measured in years since January 1st, 1950. If • There are 500 fish in the pond in 1975 • B) There are 500 more fish in 1975 than there were in 1950 • C) On average, the fish population increased by 500 per • year over the first 25 years following 1950 • D) On Jan. 1st, 1975, the fishing population was growing • at a rate of 500 fish per year • E) None of the above

  19. f(x) = position function f’(x) = velocity function f”(x) = acceleration function

  20. The end • p. 101 (1-10, 12, 16, 18, 25, 26 a-e, 28)

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