• 3, 4, 5
• 5, 12, 13
• 7, 24, 25
• 8, 15, 17

\begin{array}{ r | c | c | c | c | c } f & 0 & \frac{\pi}{6} & \frac{\pi}{4} & \frac{\pi}{3} & \frac{\pi}{2} \\ \hline \sin & 0 & \frac{1}{2} & \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{2} & 1 \\ \cos & 1 & \frac{\sqrt{3}}{2} & \frac{\sqrt{2}}{2} & \frac{1}{2} & 0 \\ \tan & 0 & \frac{1}{\sqrt{3}} & 1 & \sqrt{3} & \textrm{undef} \\ \end{array}

\[ sin ^2 x + cos ^2 x = 1 \]

\[ 1 + \tan ^2 x = \sec ^2 x \]

\[ 1 + \cot ^2 x = csc ^2 x \]

\begin{array}{ r | l } \sin^{-1} & [ -\frac{\pi}{2}, \frac{\pi}{2} ] \\ \cos^{-1} & [ 0, \pi ] \\ \tan^{-1} & ( -\frac{\pi}{2}, \frac{\pi}{2} ) \\ \csc^{-1} & [-\frac{\pi}{2}, 0) \cup (0, \frac{\pi}{2}] \\ \sec^{-1} & [0, \frac{\pi}{2}) \cup (\frac{\pi}{2}, \pi] \\ \cot^{-1} & (0, \pi) \\ \end{array}

\[ (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 \]

\[ a^2 - b^2 = (a + b)(a - b) \]

\[ a^3 + b^3 = (a + b)(a - b)^2 \]

\[ a^3 - b^3 = (a - b)(a + b)^2 \]

\[ \log a + \log b = \log(ab) \]

\[ a\log b = \log(b^a) \]

The degree of the function determines if a function is odd/even. An odd function has an infinite range while an even function "mirrors" and has a limited range.

A limit exists at a point when the function approaches a value at the point, whether or not the function exists at that point. When a left-hand and right-hand limit both exist at the same point, then a limit exists at that point.

A function is continuous when the limit exists (left-hand and right-hand) and $f(x)$ exists at that point.

\[ \lim_{x \to c} f(x) = f(c) \]

An interval or function is continuous if the domain exists for all real numbers, and a limit exists at the x-value such that $x$ is any arbitrarily selected number.

A continuous function must be continuous at all points $(x, f(x))$ in its domain.

• Jump discontinuity
• Infinite discontinuity
• Removable discontinuity

Sum rule: $(f(x) + g(x))' = f'(x) + g'(x)$

Product rule: $(f(x) \cdot g(x))' = f(x)g'(x) + f'(x)g(x)$

Quotient rule: $(\frac{f(x)}{g(x)})' = \frac{f'(x)g(x) - f(x)g'(x)}{g^2(x)}$

Reciprocal rule: $(\frac{1}{f(x)})' = -\frac{f'(x)}{f^2(x)}$

Chain rule: $(f(g(x)))' = f'(g(x)) \cdot g'(x)$

$\log_b'(a) = \frac{1}{a \ln b}$

$(b^a)' = b^a \ln b$

$\ln'(a) = \frac{1}{a}$

$(e^a)' = e^a$

$(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}$

\[ \sin(x)' = \cos(x) \]

\[ \cos(x)' = -\sin(x) \]

\[ \tan(x)' = \sec^2(x) \]

\[ \csc(x)' = -\cot x \csc x \]

\[ \sec(x)' = \sec x \tan x \]

\[ \cot(x)' = -\csc^2(x) \]

$\arcsin'(x) = \frac{1}{\sqrt{1 - x^2}}$

$\arccos'(x) = -\frac{1}{\sqrt(1 - x^2)}$

$\arctan'(x) = \frac{1}{x^2 + 1}$

$(cot^{-1})'(x) = -\frac{1}{x^2 + 1}$

$(csc^{-1})'(x) = -\frac{1}{|x|\sqrt{x^2 - 1}}$

$(sec^{-1})'(x) = \frac{1}{|x|\sqrt{x^2 - 1}}$

\[ \lim_{x \to 0} \sin(\frac{a \sin x}{b}) = \frac{a}{b} \]

\begin{array}{ c c c c } f(x) & \cup & \cap \\ f'(x) & \textrm{inc} & \textrm{dec} \\ f''(x) & + & - \end{array}

$f'(c) < 0$ max

$f'(c) > 0$ min

A linear approx is an overestimate if $f(x)$ is concave down. A linear approx is an underestimate if $f(x)$ is concave up.

\[ \int_{a}^{b} f(x) dx = - \int_{b}^{a} f(x) dx \]

\[ \int_{a}^{b} f(x) dx + \int_{b}^{c} f(x) dx = \int_{a}^{c} f(x) dx \]

\[ \frac{f(b) - f(a)}{b - a} \]

\[ \frac{1}{b - a} \int_{a}^{b} f(x) dx \]

\[ \frac{s(b) - s(a)}{b - a} \]

If $f(x)$ is continuous on $[a, b]$, then $\frac{d}{dx} \int_a^x f(t)dt = f(x)$.

If $f(x)$ is continuous on $[a, b]$ and $F(x)$ is an antiderivative of $f(x)$, then $\int_a^b f(x) = F(b) - F(a)$

If $f'(x) > 0$ for all $x$, then RRAM would be an overestimate, and LRAM would be an underestimate.

If $f'(x) < 0$ for all $x$, then RRAM would be an underestimate, and RRAM would be an overestimate.

\[ \frac{dy}{dt} = ky \Rightarrow y = y_0e^{kt} \]

where $y_0 = C$

$k < 0$ exp. decay

$k > 0$ exp. growth

\[ \frac{dy}{dt} = ky(L - y) \Rightarrow y = \frac{L}{1 + Ae^{-Lkt}} \]

\[ \lim_{t \to \infty} y = L \]

Point of inflection at $\frac{L}{2}$, where $y$ is growing the fastest

\[ A = \pi r^2 \]

\[ V = \pi \int (r(x))^2 dx \]

\[ A = \pi(R^2 - r^2) \]

\[ V = \pi \int (R(x)^2 - r(x)^2) dx \]

\[ L = \int_a^b \sqrt{1 + (\frac{dy}{dx})^2} dx \]

Depending on the problem, $\frac{dx}{dy}$ may be used instead.

\[ \frac{dy}{dx} = \frac{dy}{dt} \cdot \frac{dt}{dx} \]

\[ \frac{d^2y}{dx^2} = \frac{d}{dt} \frac{dy}{dx} \cdot \frac{dt}{dx} \]

\[ x = r cos \theta \]

\[ y = r sin \theta \]

\[ \frac{dy}{dx} = \frac{\frac{dy}{d \theta}}{\frac{dx}{d \theta}} \]