Cheatsheet

Limits

Trigonometric Limits

\[ \begin{align} \lim_{x \to 0} \frac{\sin ax}{ax} &= 1 \\\\ \lim_{x \to 0} \frac{1 - \cos ax}{ax} &= 0 \\\\ \end{align} \]

L'Hopital's Rule

When evaluating a limit to an indeterminate form, you can use L'Hopital's Rule to evaluate the limit:

\[ \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} \quad \text{or} \quad \lim_{x \to \infty} \frac{f(x)}{g(x)} = \lim_{x \to \infty} \frac{f'(x)}{g'(x)} \]

Intermediate Value Theorem

If \(f\) is continuous on \([a, b]\) and \(N\) is between \(f(a)\) and \(f(b)\), then there exists a number \(c\) in \([a, b]\) such that \(f(c) = N\).

Mean Value Theorem

If \(f\) is continuous on \([a, b]\) and differentiable on \((a, b)\), then there exists a number \(c\) in \((a, b)\) such that:

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

Derivatives

Trigonometric Derivatives

\[ \begin{align} \frac{d}{dx} \sin x &= \cos x \\\\ \frac{d}{dx} \cos x &= -\sin x \\\\ \frac{d}{dx} \tan x &= \sec^2 x \\\\ \frac{d}{dx} \sec x &= \sec x \tan x \\\\ \frac{d}{dx} \csc x &= -\csc x \cot x \\\\ \frac{d}{dx} \cot x &= -\csc^2 x \\\\ \end{align} \]

Inverse Trigonometric Derivatives

\[ \begin{align} \frac{d}{dx} \sin^{-1} x &= \frac{1}{\sqrt{1 - x^2}} \\\\ \frac{d}{dx} \cos^{-1} x &= -\frac{1}{\sqrt{1 - x^2}} \\\\ \frac{d}{dx} \tan^{-1} x &= \frac{1}{1 + x^2} \\\\ \frac{d}{dx} \sec^{-1} x &= \frac{1}{|x|\sqrt{x^2 - 1}} \\\\ \frac{d}{dx} \csc^{-1} x &= -\frac{1}{|x|\sqrt{x^2 - 1}} \\\\ \frac{d}{dx} \cot^{-1} x &= -\frac{1}{1 + x^2} \\\\ \end{align} \]

Inverse Function

If \(f\) is a one-to-one function, then the inverse function \(f^{-1}\) has the derivative:

\[ \frac{d}{dx} f^{-1}(x) = \frac{1}{f'(f^{-1}(x))} \]

Integrals

Fundamental Theorem of Calculus

If \(f\) is continuous on \([a, b]\), then:

\[ \int_a^b f(x) \, dx = F(b) - F(a) \]

where \(F\) is an antiderivative of \(f\).

Average Value of a Function

The average value of a function \(f\) on \([a, b]\) is:

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

Parametric Equations

Slope of a Parametric Curve

The slope of a parametric curve \(x = f(t)\), \(y = g(t)\) at a point \((x, y)\) is:

\[ \frac{dy}{dx} = \frac{g'(t)}{f'(t)} \quad \text{where} \quad \frac{dx}{dt} \neq 0 \]

Velocity of a Parametric Curve

The velocity of a parametric curve \(x = f(t)\), \(y = g(t)\) at a point \((x, y)\) is:

\[ \sqrt{[f'(t)]^2 + [g'(t)]^2} \]

Velocity Vector of a Parametric Curve

The velocity vector of a parametric curve \(x = f(t)\), \(y = g(t)\) is:

\[ \langle f'(t), g'(t) \rangle \]

Arc Length of a Parametric Curve

The arc length of a parametric curve \(x = f(t)\), \(y = g(t)\) from \(t = a\) to \(t = b\) is:

\[ \int_a^b \sqrt{[f'(t)]^2 + [g'(t)]^2} \, dt \]