Math 100 Final - Linear approximations, Taylor Polynomials and Maclaurin Polynomials

Hell

"What is the formula for the Linear Approximation $L(x)$ of $f(x)$ at $x=a$?"

"$L(x) = f(a) + f'(a)(x-a)$"

"What is the difference between a Taylor Polynomial and a Maclaurin Polynomial?"

"A Taylor Polynomial can be centered at any point $x=a$

"What is the general formula for the coefficient of the $(x-a)^k$ term in a Taylor Polynomial?"

"$\frac{f^{(k)}(a)}{k!}$"

"What is the Maclaurin polynomial for $e^x$?"

"$1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots + \frac{x^n}{n!}$"

"What is the Maclaurin polynomial for $\sin(x)$?"

"$x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \dots$ (Odd powers

"What is the Maclaurin polynomial for $\cos(x)$?"

"$1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \dots$ (Even powers

"What is the Maclaurin polynomial for $\frac{1}{1-x}$ (Geometric Series)?"

"$1 + x + x^2 + x^3 + \dots + x^n$"

"What is the Maclaurin polynomial for $\ln(1+x)$?"

"$x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \dots$ (Alternating signs

"What is the Maclaurin polynomial for $\arctan(x)$?"

"$x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \dots$ (Odd powers

"When should you use Linear Approximation?"

"Use it for quick

"When should you use a higher-degree Taylor Polynomial instead of a Linear Approximation?"

"Use a Taylor Polynomial (degree $n \ge 2$) when a straight line is not accurate enough because the function curves significantly near the point of interest

"When should you use a Maclaurin Polynomial specifically?"

"Use it when you are approximating values very close to $x=0$

"How do you decide the 'center' ($a$) for a Taylor Polynomial approximation?"

"Choose an $a$ that is close to the value you are trying to estimate

"Which approximation method is best for evaluating limits like $\lim_{x \to 0} \frac{\sin x - x}{x^3}$?"

"Maclaurin Polynomials. Replacing functions with their series expansions (e.g.