Glossary

Arc Notation

Criticality: 1

An alternative naming convention for inverse trigonometric functions, where 'arc' precedes the trigonometric function name (e.g., arcsin, arccos, arctan).

Example:

Instead of writing $\sin^{-1}(0.5)$ , you can use arc notation and write $\arcsin(0.5)$ to represent the angle whose sine is 0.5.

Chain Rule

Criticality: 3

A fundamental rule in calculus used for differentiating composite functions, stating that the derivative of $f(g(x))$ is $f'(g(x)) \cdot g'(x)$.

Example:

To find the derivative of $y = \cos(e^{2x})$ , you must apply the chain rule, treating $e^{2x}$ as the inner function.

Chain Rule

Criticality: 3

A fundamental rule for differentiating composite functions, stating that the derivative of $ f(g(x)) $ is found by multiplying the derivative of the outer function (evaluated at the inner function) by the derivative of the inner function.

Example:

To find the derivative of $y = (4x - 7)^5$ , you apply the chain rule to get $5(4x - 7)^4 \cdot 4$ .

Chain Rule

Criticality: 3

A fundamental rule for differentiating composite functions. It states that the derivative of $ f(g(x)) $ is $ f'(g(x)) \cdot g'(x) $.

Example:

To find the derivative of $\sin(x^2)$ , you apply the Chain Rule: $\cos(x^2) \cdot 2x$ .

Composite Function

Criticality: 3

A function formed by combining two functions, where the output of one function becomes the input of another, typically written as $ f(g(x)) $.

Example:

If $f(x) = \sqrt{x}$ and $g(x) = x^2 + 1$ , then $f(g(x)) = \sqrt{x^2 + 1}$ is a composite function.

Concavity

Criticality: 3

Concavity describes the direction in which the graph of a function is curving. A function can be concave up (curving upwards) or concave down (curving downwards).

Example:

When analyzing the profit growth of a company, if the second derivative of the profit function is positive, it indicates that the profit growth is accelerating, meaning the profit curve exhibits concavity upwards.

Derivative

Criticality: 3

A measure of how a function's output changes with respect to its input, representing the instantaneous rate of change or the slope of the tangent line at any given point.

Example:

The derivative of $f(x) = x^3$ is $f'(x) = 3x^2$ , which tells you the slope of the tangent line at any point on the curve.

Derivative of Inverse Cosine

Criticality: 3

The rate of change of the inverse cosine function, $\cos^{-1} x$, with respect to its input, given by the formula $-\frac{1}{\sqrt{1 - x^2}}$.

Example:

If you're modeling the angle of elevation of a camera tracking a horizontally moving object, the derivative of inverse cosine could be relevant.

Derivative of Inverse Sine

Criticality: 3

The rate of change of the inverse sine function, $\sin^{-1} x$, with respect to its input, given by the formula $\frac{1}{\sqrt{1 - x^2}}$.

Example:

When analyzing the rate at which an angle changes in a right triangle as the opposite side varies, you might need to apply the derivative of inverse sine.

Derivative of Inverse Tangent

Criticality: 3

The rate of change of the inverse tangent function, $ an^{-1} x$, with respect to its input, given by the formula $\frac{1}{1 + x^2}$.

Example:

In physics, when calculating the rate of change of an angle of incidence as a light ray passes through a medium, the derivative of inverse tangent often appears.

Derivatives

Criticality: 3

A measure of how a function changes as its input changes, representing the instantaneous rate of change or the slope of the tangent line to the function's graph at a given point.

Example:

If a car's position is given by $s(t) = t^2$ , its derivative, $s'(t) = 2t$ , tells you its instantaneous speed at any time $t$ .

Domain Restrictions

Criticality: 2

Specific ranges of input values for which a function or its derivative is mathematically defined, crucial for inverse functions to be well-behaved.

Example:

Understanding the domain restrictions for $\sin^{-1} x$ (i.e., $x \in [-1, 1]$ ) is vital because its derivative is only valid for $x \in (-1, 1)$ .

Higher-Order Derivative

Criticality: 3

Higher-order derivatives are derivatives of a function taken more than twice (e.g., third, fourth, or nth derivatives). They are obtained by repeatedly differentiating the function.

Example:

To understand the subtle nuances of a complex polynomial function's behavior, an engineer might calculate its higher-order derivatives to find specific points of curvature or inflection.

Inside Function

Criticality: 2

In a composite function $ f(g(x)) $, the *inside function* is $ g(x) $, which is the first function evaluated and whose output serves as the input for the outer function.

Example:

For the function $y = e^{\sin(x)}$ , the expression $\sin(x)$ is the inside function.

Inverse Function Theorem

Criticality: 3

A theorem that provides a method to find the derivative of an inverse function by relating it to the derivative of the original function.

Example:

If you know the derivative of $f(x) = x^3$ , the inverse function theorem allows you to easily find the derivative of its inverse, $g(x) = \sqrt[3]{x}$ .

Inverse Function Theorem

Criticality: 3

A theorem that provides a formula for the derivative of the inverse of a function, stating that $ (f^{-1})'(a) = \frac{1}{f'(f^{-1}(a))} $.

Example:

To find the derivative of $\ln(x)$ (which is the inverse of $e^x$ ) at a specific point, you can use the Inverse Function Theorem by evaluating $1/e^{f^{-1}(x)}$ .

Inverse Functions

Criticality: 3

A function that 'reverses' the action of another function. If $ f(x) = y $, then its inverse, $ f^{-1}(y) = x $, maps $ y $ back to $ x $.

Example:

If $f(x) = 2x$ doubles a number, its inverse function, $f^{-1}(x) = x/2$ , halves it, bringing you back to the original number.

Inverse Trigonometric Functions

Criticality: 3

Functions that are the inverses of the basic trigonometric functions (sine, cosine, tangent, etc.), used to determine angles from given ratios.

Example:

If you know the cosine of an angle is 0.8, you use an inverse trigonometric function like $\arccos(0.8)$ to find the angle itself.

Jerk

Criticality: 1

Jerk is the third derivative of position with respect to time, representing the rate of change of acceleration. It describes how smoothly or abruptly acceleration changes.

Example:

In roller coaster design, minimizing jerk is crucial to ensure a comfortable ride, as sudden changes in acceleration can cause discomfort or injury to passengers.

L'Hopital's Rule

Criticality: 2

L'Hopital's Rule is a method used to evaluate limits of indeterminate forms (like 0/0 or ∞/∞) by taking the derivatives of the numerator and denominator.

Example:

When trying to find the limit of a complex rational function as x approaches a certain value, if direct substitution yields an indeterminate form, applying L'Hopital's Rule can simplify the expression and reveal the true limit.

One-to-one function

Criticality: 2

A function where each element in the domain maps to a unique element in the codomain, meaning no two distinct inputs map to the same output. This property is necessary for a function to have an inverse.

Example:

The function $f(x) = x^3$ is a one-to-one function because each unique input yields a unique output, allowing it to have an inverse. However, $f(x) = x^2$ is not, as both 2 and -2 map to 4.

Product Rule

Criticality: 2

A rule used to differentiate a function that is the product of two other differentiable functions, given by $ (uv)' = u'v + uv' $.

Example:

To differentiate $f(x) = x^3 \cos(x)$ , you would use the product rule to find $3x^2 \cos(x) - x^3 \sin(x)$ .

Quotient Rule

Criticality: 2

A rule used to differentiate a function that is the ratio of two other differentiable functions, given by $ (\frac{u}{v})' = \frac{u'v - uv'}{v^2} $.

Example:

To differentiate $f(x) = \frac{ an(x)}{x^2}$ , you would apply the quotient rule.

Reciprocal of a Derivative

Criticality: 2

The inverse of the derivative $ \frac{dy}{dx} $, expressed as $ \frac{dx}{dy} $. It represents the rate of change of the input variable with respect to the output variable.

Example:

If $\frac{dy}{dx}$ tells you how fast the volume of a balloon changes with respect to its radius, then its reciprocal of a derivative, $\frac{dx}{dy}$ , would tell you how fast the radius changes with respect to the volume.

Second Derivative

Criticality: 3

The second derivative of a function is the derivative of its first derivative. It measures the rate of change of the rate of change of the original function.

Example:

If a car's position is described by a function, its first derivative is velocity, and its second derivative represents the acceleration, indicating how quickly the velocity is changing.

Trigonometric Identities

Criticality: 2

Equations involving trigonometric functions that are true for all values of the variables for which the functions are defined.

Example:

The identity $an^2 x + 1 = \sec^2 x$ is a key trigonometric identity used to simplify expressions when deriving the derivative of inverse tangent.