Week 6 Checkpoint

$0$

$4$

Undefined (division by zero)

$2$

$f(a)$ is defined

$\lim_{x\to a} f(x)$ exists

$\lim_{x\to a} f(x) = f(a)$ (the limit exists and equals the value)

$f$ is differentiable at $a$

$\displaystyle\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$

$\dfrac{f(x+h)-f(x)}{h}$ for a fixed small $h$

$\displaystyle\lim_{h\to 0}\big(f(x+h)-f(x)\big)$

$\dfrac{f(x)}{x}$

$2x\ln x$

$2x\ln x + x$

$2$

$x$

$\frac{d}{dx}e^x=e^x$ and $\frac{d}{dx}\ln x=\frac{1}{x}$

$\frac{d}{dx}e^x=xe^{x-1}$ and $\frac{d}{dx}\ln x=\ln x$

$\frac{d}{dx}e^x=e^{x}\ln x$ and $\frac{d}{dx}\ln x=x$

$\frac{d}{dx}e^x=1$ and $\frac{d}{dx}\ln x=\frac{1}{\ln x}$

$O(h)$ — the same as the forward difference

$O(h^2)$ — more accurate for small $h$

$O(1)$ — independent of $h$

$O(1/h)$ — worse for small $h$

$1.0$

$2.0$

$2.01$

$4.0$