Our problem: trying to figure out the instantaneous ROC of f(x), graphed below, at x = 1.[br]Let's get our bearings first:[br][list][br][*]f(x) is in blue.[br][/*][*]The secant line between x = 1 and x = 1 + h is in brown. Note its slope is shown also.[br][/*][*]For some reason, the tangent line at x = 1 is shown also. It's in red. Note that you can toggle it on and off. You may want to start with it off so the drawing isn't too cluttered.[br][/*][/list][br][br]Ok, now answer some questions and follow directions below:[br][list=1][br][*] The secant line's slope represents the __________ ROC of f(x) from ___ to ___.[br][/*][*] Do you agree that the smaller h is, the better the average ROC approximates the instantaneous ROC?[br][/*][*] Let's improve our approximation by shrinking h: drag the point labeled "1 + h" to the left.[br][/*][*] Note that as h approaches 0, the secant line starts becoming a __________ line at x = 1.[br][/*][*] Here is the gigantic realization of Newton (and some others as well): if the slope of the secant line is the average ROC of f(x), then the slope of the _________ line at x = 1 is the instantaneous ROC at x = 1.[br][/*][*] Write this slope using limit notation. Hint: h is approaching 0, and the formula for slope is - what?[br][/*][/list]

If you wrote the slope of the tangent line properly, you've just written the [i]definition of a derivative[/i]. It gets the bronze medal for the most important formula in calculus. You'll learn the gold and silver later.[br][br]For now, when I ask you what the derivative of f(x) at x = 7 means, you should say, "It's the instantaneous ROC of f(x) at x = 7".[br][br]Derivative means instantaneous ROC and vice versa. On a graph, the derivative at a point is the slope of the tangent line at that point.[br][br]One last thing, the fraction part of the definition of derivative is called the [i]difference quotient[/i]. It's vocab to know, nothing more.