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PHY221 Ch0: Math & Physics preliminariesPowerPoint Presentation

PHY221 Ch0: Math & Physics preliminaries

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PHY221 Ch0: Math & Physics preliminaries

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PHY221 Ch0: Math & Physics preliminaries

Consider a function of time:

Plot it:

The rate of change of f between t1 and t2 is given by:

Rate of change=rise over run = slope of secant!

When , secant becomes tangent and

rate of change becomes derivative.

Math: Derivatives

Rate of change of a function.

PHY221 Ch0: Math & Physics preliminaries

So the derivative

Let’s compute the derivative in our example :

First, the rate of change :

Then the limit as :

Final result is:

Math: Derivatives

Derivative

PHY221 Ch0: Math & Physics preliminaries

Derivatives of common functions:

Product and composition rule:

Math: Derivatives

Derivative

PHY221 Ch0: Math & Physics preliminaries

Consider a function of time:

Plot it:

For fun, let’s compute the area under plot from, say, t=1s to t=3s (f-i)

But g(t) is also the derivative of

Let’s compute

Miracle? Consider

Math: Integrals

Integral area under curve:

PHY221 Ch0: Math & Physics preliminaries

Riemann Sum: let’s compute

For each “small enough” slice #i we have

Why useful?

Consider that we know the acceleration a of an object.

Since the goal of the game is to find velocity and position and since a=dv/dtand v=dx/dt we need integrals!

Math: Integrals

Integral area under curve:

1. Compute derivative from scratch, as we did on slide #l and 2 of the functions:

and

2. Now compute the derivative of f1, plot it, and compute the area under the graph from to .

3. Verify that it is equal to the corresponding quantity in terms of f1