Topic 2.1 Extended E â The method of slopes

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Topic 2.1 Extended E â The method of slopes - PowerPoint PPT Presentation

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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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:

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.

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.