1. The problem: You want to learn important formulas and concepts from the first chapter of Calculus and Analytical Geometry. 2. Key concepts include limits, continuity, differentiation, and basic geometry of curves. 3. Important formulas and concepts: - Limit of a function: $$\lim_{x \to a} f(x) = L$$ means as $x$ approaches $a$, $f(x)$ approaches $L$. - Continuity: A function $f$ is continuous at $x=a$ if $$\lim_{x \to a} f(x) = f(a)$$. - Derivative definition: $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$. - Basic differentiation rules: - Power rule: $$\frac{d}{dx} x^n = n x^{n-1}$$ - Sum rule: $$\frac{d}{dx} [f(x) + g(x)] = f'(x) + g'(x)$$ - Product rule: $$\frac{d}{dx} [f(x) g(x)] = f'(x) g(x) + f(x) g'(x)$$ - Quotient rule: $$\frac{d}{dx} \left( \frac{f(x)}{g(x)} \right) = \frac{f'(x) g(x) - f(x) g'(x)}{[g(x)]^2}$$ - Chain rule: $$\frac{d}{dx} f(g(x)) = f'(g(x)) g'(x)$$ - Equation of a straight line: $$y = mx + c$$ where $m$ is slope and $c$ is y-intercept. - Distance formula between points $(x_1,y_1)$ and $(x_2,y_2)$: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$. - Midpoint formula: $$\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$$. 4. These formulas form the foundation for solving problems in calculus and analytical geometry. 5. Practice applying these formulas to understand their use and meaning.