MATH 144: Calculus for the Physical Sciences I

Subsection 1.1.1 Things to know

• If $ax^2+bx+c=0\text{,}$ the roots are $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\text{;}$ the sum of the roots is $-b/a\text{;}$ the product of the roots is $c/a\text{.}$
• $\displaystyle x^{-r} = \frac{1}{x^r}.$
• $x^r x^s=x^{r+s}\text{.}$
• $x^{1/n} = \sqrt[n]{x},$ for $n$ integer. If $n$ is even, then $x$ must be non-negative and $\sqrt[n]{x}$ denotes the non-negative root of $x\text{.}$
• $\left( x^r \right)^s = x^{r \cdot s}\text{,}$ assuming that $x \geq 0\text{.}$ Note that care must be taken if $x$ is negative. For instance, $(x^2)^{1/2} \neq x$ if $x \lt 0\text{;}$ rather, $(x^2)^{1/2} = |x|\text{.}$
• Finally, avoid common mistakes: remember that, in general,
  $$\begin{gather*} \sqrt{x+y} \neq \sqrt{x} + \sqrt{y},\\ \frac{1}{x+y} \neq \frac{1}{x} + \frac{1}{y},\\ (x+y)^2 \neq x^2 + y^2. \end{gather*}$$

Subsection 1.1.2 Review videos