Notes on Calculus (Differential Calculus) and Limits

Average Rate of Change Formula

The average rate of change of a function over an interval \[t1, t2\] is calculated using:
v{avg} = \frac{s(t2) - s(t1)}{t2 - t_1}

Instantaneous Rate of Change Formula

The instantaneous rate of change at a point can be found using limits:
\lim_{h \to 0} \frac{f(x + h) - f(x)}{h}
where

  • (f(x)) is the function, and

  • (h) is the interval tending to zero.

Finding the Derivative Using First Principles

To find the derivative of a function using the limit definition:

  1. Identify the function (f(x)) that needs differentiation.

  2. Apply the limit definition:

    • f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}

  3. Simplify the expression and evaluate the limit to arrive at the derivative.

Example: Derivative of a Polynomial

For a polynomial function (f(x) = x^2 + 2), you would:

  • Calculate (f(x + h) = (x + h)^2 + 2).

  • Then simplify:
    f(x + h) = x^2 + 2xh + h^2 + 2

  • Apply the limit definition:
    f'(x) = \lim{h \to 0} \frac{(x^2 + 2xh + h^2 + 2) - (x^2 + 2)}{h} = \lim{h \to 0} \frac{2xh + h^2}{h} = \lim_{h \to 0} (2x + h) = 2x

Conclusion

Understanding these formulas and methods is essential for solving problems in calculus, particularly when dealing with rates of change and the behavior of functions.