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

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Presentation transcript:

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

Warmup:

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

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

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

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

dx does not mean d times x ! dy does not mean d times y !

does not mean !

does not mean times !

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

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. 

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

Graph the derivative of the function Enter the function and derivative. Use TABLE to help find a window.TABLE Set a window. Use the Thick option for the derivative and graph the functions.Thick

Graph of f(x) Make a table of approximations of slopes of tangent lines at the pts Pointxslope A0 B1.5 C2.5 D3 E5 F6 ? Now lets take these values and make a graph of all the slopes ( f ‘ (x) graph )

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

Sketch

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: I.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 A)I onlyB) II onlyC) III onlyD) I and II E) I and IIIF) I, II, and III

The function y = f(x) measures the fish population in Lincoln Pond at time x, where x is measured in years since January 1 st, If A)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. 1 st, 1975, the fishing population was growing at a rate of 500 fish per year E) None of the above

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

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