1. **Stating the problem:** We want to understand how Taylor's and Maclaurin's formulas are used to approximate functions using derivatives. 2. **Formula and explanation:** - The Taylor series of a function $f(x)$ centered at $a$ is given by: $$f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x - a)^n$$ where $f^{(n)}(a)$ is the $n$th derivative of $f$ evaluated at $a$. - The Maclaurin series is a special case of the Taylor series centered at $a=0$: $$f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!}x^n$$ 3. **Important rules:** - The more terms we include, the better the approximation near $a$. - Derivatives give the coefficients, capturing the function's behavior. 4. **Intermediate work example:** Approximate $e^x$ near $0$ using Maclaurin series up to the 3rd degree: - $f(x) = e^x$, so $f^{(n)}(x) = e^x$ for all $n$. - Evaluate derivatives at $0$: $f^{(n)}(0) = 1$. - Series up to $n=3$: $$e^x \approx 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} = 1 + x + \frac{x^2}{2} + \frac{x^3}{6}$$ 5. **Explanation:** This polynomial approximates $e^x$ near $0$. The derivatives determine the shape and curvature, making the approximation accurate close to the center point. **Final answer:** Taylor and Maclaurin formulas use derivatives to build polynomial approximations of functions near a point, enabling easier calculations and insights into function behavior.