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 Extended E – The method of slopes.
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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.
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 (16461716).
The fourstep process of taking the derivative is outlined above in the second form:
The whole process will be represented with the new symbol dx/dt:
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
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.
v
t
t
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.