Topic 2.1 Extended E – The method of slopes

1. Instantaneous Velocity FYI: Think of x as a crude difference in x's. Think of dx as a very fine difference in x's. Think of t as a crude difference in t's. Think of dt as a very fine difference in t's. Topic 2.1 ExtendedE – The method of slopes FYI: Leibniz invented calculus about a decade after Newton, apparently without having seen much of Newton's work on the subject. Leibniz' notation is still used today. Newton's is obscure and ignored. We have two equivalent definitions of instantaneous velocity, both of which are hard to use: x t dx dt limit t→0 = v = x(t + t) - x(t) t dx dt limit t→0 v = = Before we show yet another way to find the derivative we introduce a new notation, courtesy of Gottfried Wilhelm Leibniz (1646-1716). The four-step process of taking the derivative is outlined above in the second form: The whole process will be represented with the new symbol dx/dt:

2. x x(t) x t x t x t x t dx dt x(t + t) x(t + t) x(t + t) x(t + t) t x(t) t t+t t+t t+t t+t Topic 2.1 ExtendedE – The method of slopes Recall: Graphically, as t→0, the average velocity becomes the instantaneous velocity. The method of slopes is another way to get the velocity function from the position function. Step 1: Find the slopes of various tangents, and plot them in a new graph. Step 2: If possible, identify the graph with a function.

3. x v t t Topic 2.1 ExtendedE – The method of slopes Here is a sample problem: o + - o + - + o A particle moves along the x axis with x(t) shown in the figure. Make a rough sketch of velocity vs. time for this motion. Velocity is the SLOPE of the x vs. t graph... Of course, the more accurate our slopes, the more accurate our graph of v vs. t. FYI: Neither graph is easily identifiable as a function. Sometimes all we need is a ROUGH SKETCH.