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:
$v<em>{avg} = \frac{s(t</em>2) - s(t<em>1)}{t</em>2 - t_1}$

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:
  $f'(x) = \lim<em>{h \to 0} \frac{(x^2 + 2xh + h^2 + 2) - (x^2 + 2)}{h} = \lim</em>{h \to 0} \frac{2xh + h^2}{h} = \lim_{h \to 0} (2x + h) = 2x$

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.