August 26 - Calculus Notes: Derivatives, Integrals, and Functions

Rates of Change and the Tangent Problem

• Calculus centers on two ideas: rates of change (derivatives) and areas under curves (integrals).

Derivatives: Instantaneous Rate of Change

• Derivatives describe how a quantity changes at a specific moment.

• Examples: Velocity (instantaneous rate of change of position) and acceleration (rate of change of velocity).

  • Velocity $v(t) = ds/dt$; Acceleration $a(t) = dv/dt = d^2s/dt^2$.

• Geometric Intuition: The slope of a curve at a point (the tangent line) represents the instantaneous rate of change.

• Formal Definition (Limit): The derivative $s'(t)$ is the limit of the average rate of change as the interval shrinks to zero.

  • Average Rate of Change: $\frac{\Delta s}{\Delta t} = \frac{s(t+h) - s(t)}{h}$.

  • Instantaneous Rate of Change (Derivative): $s'(t) = \lim_{h\to 0}\frac{s(t+h) - s(t)}{h}$.

• Tangent Line Equation: Given $f'(x<em>0)$ and $f(x</em>0)$, the tangent line at $x = x<em>0$ is $y = f'(x</em>0)(x - x<em>0) + f(x</em>0)$.

Integrals: Area Under a Curve

• Integrals find the area between a curve $y = f(x)$ and the x-axis.

• For simple shapes, geometric formulas suffice (e.g., triangle area for $y=x$ from $x=0$ to $x=1$ is $\frac{1}{2}$).

• General Method (Riemann Sums): For complex curves, approximate the area using rectangles (Riemann sums).

  • Partition interval $[a, b]$ into $n$ subintervals of width $\Delta x = \frac{b-a}{n}$.

  • Approximate Area: $\text{Area approx} = \sum<em>{i=1}^{n} f(x</em>i^*) \Delta x$.

  • Exact Area (Limit): As $n \to \infty$, this sum becomes the definite integral: $A = \int_{a}^{b} f(x) \, dx$.

Continuity

• Most calculus concepts rely on continuous functions; piecewise continuity is also common.

• Piecewise Continuous Function Example: The absolute value function:

  • $|x| = \begin{cases} x, &amp; x \ge 0 \ -x, &amp; x &lt; 0 \end{cases}$