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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 . .
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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:
Ch 0B: Assignments 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
