\[\sinh x = \frac{e^x - e^{-x}}{2}\] \[\cosh x = \frac{e^x + e^{-x}}{2}\] \[\tanh x = \frac{\sinh x}{\cosh x} = \frac{e^x - e^{-x}}{e^x + e^{-x}}\]
Inverse Hyperbolic Functions
\[\sinh^{-1} x = \ln(x + \sqrt{x^2 + 1})\] \[\cosh^{-1} x = \ln(x + \sqrt{x^2 - 1}), \quad x \geq 1\] \[\tanh^{-1} x = \frac{1}{2}\ln\left(\frac{1+x}{1-x}\right), \quad |x| < 1\]
Important Identities
\[\cosh^2 x - \sinh^2 x = 1\] \[1 - \tanh^2 x = \text{sech}^2 x\]
Graphs of hyperbolic sine, cosine, and tangent functions. Note that $\sinh x$ is odd, $\cosh x$ is even, and $\tanh x$ has horizontal asymptotes at $y = ±1$.
If $\lim_{x \to a} \frac{f(x)}{g(x)}$ is of form $\frac{0}{0}$ or $\frac{\infty}{\infty}$
Then $\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f’(x)}{g’(x)}$
Visualization of the limit $\lim_{x \to 0} \frac{\sin x}{x} = 1$. The function approaches 1 as x approaches 0 from either direction.
Visualization of the limit $\lim_{x \to 0} \frac{e^x - 1}{x} = 1$. This fundamental limit is used in calculus to derive the derivative of e^x.
3. Function Properties
Even Functions
$f(-x) = f(x)$
Odd Functions
$f(-x) = -f(x)$
Visualization of even functions (symmetric about y-axis) and odd functions (symmetric about origin). Even functions like $x^2$ have $f(-x) = f(x)$, while odd functions like $x^3$ have $f(-x) = -f(x)$.
Inverse Functions
If $y = f(x)$, then $x = f^{-1}(y)$
$f(f^{-1}(x)) = x$
$f^{-1}(f(x)) = x$
Domain of $f^{-1}$ = Range of $f$
Range of $f^{-1}$ = Domain of $f$
Demonstration of inverse functions: $f(x) = x^2$ and $f^{-1}(x) = \sqrt{x}$. The bottom graph shows that applying $f$ after $f^{-1}$ yields the identity function $f(f^{-1}(x)) = x$.
Graphs of exponential function $e^x$ and natural logarithm $\ln(x)$. Note that $e^x$ grows rapidly for positive x, while $\ln(x)$ grows very slowly. These functions are inverses of each other.
5. Continuity
A function $f$ is continuous at a point $x_0$ if:
$\lim_{x \to x_0^-} f(x)$ exists
$\lim_{x \to x_0^+} f(x)$ exists
Both limits equal $f(x_0)$
Visualization of a removable discontinuity in $f(x) = \frac{\sin x}{x}$ at $x = 0$. The limit exists and equals 1, but the function value is undefined at x = 0.
Example of a jump discontinuity where the left and right limits exist but are not equal, making the function discontinuous at that point.
Demonstration of an infinite discontinuity in $f(x) = \frac{1}{x}$ at $x = 0$, where the function values approach infinity as x approaches 0 from either side.
6. Trigonometric Functions and Identities
Basic Identities
\[\sin^2 x + \cos^2 x = 1\] \[\cot x = \frac{\cos x}{\sin x}\] \[\sin(-x) = -\sin(x)\] \[\cos(-x) = \cos(x)\]
Addition and Subtraction Formulas
\[\sin(x \pm y) = \sin x \cos y \pm \cos x \sin y\] \[\cos(x \pm y) = \cos x \cos y \mp \sin x \sin y\] \[\tan(x \pm y) = \frac{\tan x \pm \tan y}{1 \mp \tan x \tan y}\]
Double Angle Formulas
\[\sin(2x) = 2\sin x \cos x\] \[\cos(2x) = \cos^2 x - \sin^2 x = 2\cos^2 x - 1 = 1 - 2\sin^2 x\] \[\tan(2x) = \frac{2\tan x}{1 - \tan^2 x}\]
Graphs of sine, cosine, and tangent functions. Note the periodicity of these functions and the vertical asymptotes of tangent at $x = (2n+1)\frac{\pi}{2}$ where n is an integer.
Graphs of arctangent function $\tan^{-1}(x)$ and the composite function $\tan^{-1}(\sin x)$. The arctangent function maps all real numbers to the range $(-\frac{\pi}{2}, \frac{\pi}{2})$.
7. Function Composition
Function composition involves applying one function to the output of another function. If $f$ and $g$ are functions, then $(f \circ g)(x) = f(g(x))$.
Step-by-step visualization of the composition $\tan^{-1}(\sin x)$. The first function $\sin x$ maps x to $[-1, 1]$, then $\tan^{-1}$ maps this result to $[-\frac{\pi}{2}, \frac{\pi}{2}]$.
