Glossary

Antiderivatives

Criticality: 3

A function whose derivative is the given function. It represents the reverse process of differentiation.

Example:

If the derivative of $x^3$ is $3x^2$ , then $x^3$ is an antiderivative of $3x^2$ .

Basic Differentiation

Criticality: 3

Fundamental rules for finding derivatives of simple functions, including powers of x, sums, differences, and constant multiples of terms.

Example:

Applying basic differentiation to $f(x) = 4x^3 - 2x + 7$ yields $f'(x) = 12x^2 - 2$ .

Chain Rule

Criticality: 3

A rule used to find the derivative of composite functions, where one function is nested inside another, expressed as $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$.

Example:

To find the derivative of $y = (3x^2 + 1)^5$ , you apply the chain rule by differentiating the outer power function and then multiplying by the derivative of the inner polynomial.

Completing the Square

Criticality: 2

An algebraic technique used to rewrite a quadratic expression into the form $(x+a)^2 + b$ or $a(x+b)^2 + c$, often to transform an integral into a recognizable standard form.

Example:

To integrate $\int \frac{1}{x^2 + 4x + 5} , dx$ , you would use completing the square to rewrite the denominator as $(x+2)^2 + 1$ .

Composite Functions

Criticality: 3

A function formed by applying one function to the results of another function. In integration, these often require specific techniques like the reverse chain rule or substitution.

Example:

$\sin(x^2)$ is a composite function where $x^2$ is the inner function and $\sin(u)$ is the outer function.

Constant of Integration

Criticality: 2

An arbitrary constant, denoted by 'C', added to the result of an indefinite integral because the derivative of a constant is zero, meaning there are infinitely many antiderivatives for a given function.

Example:

When integrating $\int 2x , dx$ , the result is $x^2 + [object Object]$ , where C represents the constant of integration.

Continuous Function

Criticality: 2

A function whose graph can be drawn without lifting the pen, meaning it has no breaks, jumps, or holes at any point in its domain.

Example:

A polynomial function like f(x) = x^3 - 2x + 1 is a continuous function everywhere, so its limit at any point can be found by direct substitution.

Definite Integral (as a limit definition)

Criticality: 3

The signed area between a function's graph and the x-axis over a given interval, formally defined as the limit of Riemann sums as the width of the subintervals approaches zero.

Example:

To calculate the exact area under the curve of f(x) = x^2 from x=0 to x=1, one would use the definite integral, which is the limit of a sum of rectangles.

Definite Integrals

Criticality: 3

An integral evaluated over a specific interval $[a, b]$, representing the net accumulated change of a function or the signed area under its curve between the given limits.

Example:

$\int_0^1 x^2 , dx$ is a definite integral that calculates the area under $y=x^2$ from $x=0$ to $x=1$ .

Derivative

Criticality: 3

The derivative of a function represents the instantaneous rate at which the function's value changes as its input changes. It is formally defined by a limit.

Example:

If a car's position is given by $s(t)$ , its derivative $s'(t)$ gives the car's instantaneous velocity at time $t$ .

Derivative (as a limit definition)

Criticality: 3

The instantaneous rate of change of a function at a specific point, formally defined as the limit of the difference quotient as the change in the independent variable approaches zero.

Example:

The slope of the tangent line to a curve at a point is given by the derivative, which is found by evaluating the limit of [f(x+h) - f(x)]/h as h approaches 0.

Differentiating Exponentials

Criticality: 3

The specific rules for finding the derivatives of exponential functions, such as $e^{kx}$ and $a^{kx}$.

Example:

To model the growth rate of a bacterial colony described by $P(t) = 100e^{0.5t}$ , you would use the rules for differentiating exponentials to find $P'(t) = 50e^{0.5t}$ .

Differentiating Inverse Trigonometric Functions

Criticality: 2

The specific rules for finding the derivatives of inverse trigonometric functions such as arcsin, arccos, and arctan.

Example:

When calculating the rate of change of an angle given its tangent, you would use the rules for differentiating inverse trigonometric functions, such as $\arctan(x)$ .

Differentiating Logarithms

Criticality: 3

The specific rules for finding the derivatives of logarithmic functions, such as $\ln(kx)$.

Example:

When analyzing the intensity of sound, which often involves logarithmic scales, you might need to use the rules for differentiating logarithms to find the rate of change of sound intensity.

Differentiating Trigonometric Functions

Criticality: 3

The specific rules for finding the derivatives of trigonometric functions like sine, cosine, and tangent.

Example:

To determine the velocity of a pendulum swinging with displacement $heta(t) = \sin(2t)$ , you would apply the rules for differentiating trigonometric functions to get $heta'(t) = 2\cos(2t)$ .

Estimating Derivatives

Criticality: 2

The process of approximating the derivative of a function at a specific point, typically by calculating the slope of line segments between nearby points on a graph or from a table of values.

Example:

When analyzing climate data, you might estimate the derivative of global temperature over a decade by finding the average rate of change between consecutive years.

Fundamental Theorem of Calculus

Criticality: 3

A theorem that links the concepts of differentiating a function and integrating a function, providing a method for evaluating definite integrals using antiderivatives.

Example:

Using the Fundamental Theorem of Calculus, we can evaluate $\int_a^b f(x) , dx$ by finding an antiderivative $F(x)$ and calculating $F(b) - F(a)$ .

Graph

Criticality: 1

A visual representation of a function used to estimate a limit by observing the y-value the function approaches as x gets closer to a specific point.

Example:

By examining the graph of 1/x, you can see that as x approaches infinity, the function approaches 0.

Implicit Differentiation

Criticality: 3

A technique used to find the derivative of a function defined by an equation where one variable is not explicitly expressed as a function of the other, requiring the application of the chain rule for terms involving the dependent variable.

Example:

To find the slope of the tangent line to a circle defined by $x^2 + y^2 = 25$ at any point, you would use implicit differentiation.

Indefinite Integral

Criticality: 3

The integral of a function without specified limits of integration, representing the family of all possible antiderivatives of the function.

Example:

$\int \sin x , dx = -\cos x + C$ is an indefinite integral, yielding a general solution.

Inverse Function Theorem

Criticality: 2

A theorem that provides a formula for finding the derivative of the inverse of a function at a specific point, given by $(f^{-1})'(a) = \frac{1}{f'(f^{-1}(a))}$.

Example:

If you know the derivative of $f(x) = x^3 + x$ , you can use the inverse function theorem to find the derivative of its inverse function at a particular point without explicitly finding the inverse.

Limit

Criticality: 3

The value that a function approaches as its input approaches a certain value. It describes the behavior of a function near a point, not necessarily at the point itself.

Example:

As x approaches 2, the function f(x) = x + 1 approaches 3, so the limit of f(x) as x approaches 2 is 3.

Multiplying by a Conjugate (Method)

Criticality: 2

An analytical technique primarily used for limits involving square roots in the numerator or denominator, where multiplying by the conjugate helps eliminate the radical and simplify the expression.

Example:

When evaluating the limit of (sqrt(x+4) - 2)/x as x approaches 0, multiplying by a conjugate (sqrt(x+4) + 2) helps rationalize the numerator.

Multiplying by a Reciprocal (Method)

Criticality: 2

An analytical technique often used for limits of rational functions as x approaches infinity, where multiplying by the reciprocal of the highest power of x in the denominator helps determine the limit.

Example:

To find the limit of (3x^2 + 1)/(x^2 - 5) as x approaches infinity, you can divide both numerator and denominator by x^2, which is equivalent to multiplying by a reciprocal of 1/x^2.

Polynomial Long Division

Criticality: 2

An algebraic method used to divide polynomials, often applied to rational functions where the degree of the numerator is greater than or equal to the degree of the denominator, to simplify the integrand.

Example:

When integrating $\int \frac{x^3 + 1}{x - 1} , dx$ , performing polynomial long division first yields $x^2 + x + 1 + \frac{2}{x-1}$ , which is easier to integrate.

Product Rule

Criticality: 3

A rule used to find the derivative of a function that is the product of two differentiable functions, $f(x) = g(x) \cdot h(x)$, given by $f'(x) = g'(x)h(x) + g(x)h'(x)$.

Example:

To find the rate of change of $y = x^2 \sin(x)$ , you must apply the product rule because it's a multiplication of two distinct functions.

Properties of Limits

Criticality: 2

Rules that allow for the evaluation of complex limits by breaking them down into simpler components, such as the sum, difference, product, quotient, and constant multiple rules.

Example:

Using the properties of limits, you can determine that the limit of (x^2 + 3x) as x approaches 1 is the limit of x^2 plus the limit of 3x.

Quotient Rule

Criticality: 3

A rule used to find the derivative of a function that is the ratio of two differentiable functions, $f(x) = \frac{g(x)}{h(x)}$, given by $f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2}$.

Example:

When differentiating a rational function like $y = \frac{e^x}{\cos(x)}$ , the quotient rule is essential.

Reverse Chain Rule

Criticality: 3

An integration technique used for composite functions where the integrand is recognized as the result of differentiating a function using the chain rule.

Example:

To integrate $\int 2x \cos(x^2) , dx$ , you can use the reverse chain rule to see it's $\sin(x^2) + C$ .

Riemann Sums

Criticality: 2

An approximation method for calculating the definite integral of a function by dividing the area under the curve into a series of rectangles and summing their areas.

Example:

Approximating the area under $y=x^2$ from $x=0$ to $x=2$ using four rectangles of equal width is an application of Riemann sums.

Simplification

Criticality: 2

The process of rewriting an integrand into a simpler form, often using algebraic manipulation, to make it easier to integrate.

Example:

Rewriting $\frac{x^4 + x^2}{x^2}$ as $x^2 + 1$ before integrating is an act of simplification.

Simplifying Algebraically (Method)

Criticality: 3

An analytical technique used when direct substitution results in an indeterminate form (e.g., 0/0), involving factoring and canceling common terms in the numerator and denominator.

Example:

To evaluate the limit of (x^2 - 9)/(x - 3) as x approaches 3, you can factor the numerator to (x-3)(x+3) and then cancel (x-3), simplifying algebraically to x+3.

Squeeze Theorem

Criticality: 2

A theorem stating that if a function is bounded between two other functions that converge to the same limit at a certain point, then the function itself must also converge to that same limit.

Example:

If you have a function f(x) such that -x^2 <= f(x) <= x^2, then by the Squeeze Theorem, as x approaches 0, f(x) must also approach 0.

Standard Results

Criticality: 2

Pre-established integration formulas for common functions or forms that can be directly applied.

Example:

Knowing that $\int \frac{1}{x} , dx = \ln|x| + C$ is a standard result that saves time in calculations.

Substitution (Method)

Criticality: 3

An analytical method for evaluating limits of continuous functions by directly plugging the approaching value into the function.

Example:

For the function f(x) = x^2 + 5, the limit as x approaches 2 can be found by direct substitution, yielding 2^2 + 5 = 9.

Table of Values

Criticality: 1

A method for estimating a limit by evaluating the function at input values progressively closer to the point of interest from both sides.

Example:

To estimate the limit of sin(x)/x as x approaches 0, one could create a table of values for x = ±0.1, ±0.01, ±0.001.

Tangent

Criticality: 1

A straight line that touches a curve at a single point, sharing the same slope as the curve at that specific point.

Example:

The derivative of a function at $x=a$ gives the slope of the tangent line to the function's graph at the point $(a, f(a))$ .

Trapezoidal Sums

Criticality: 2

An approximation method for calculating the definite integral of a function by dividing the area under the curve into a series of trapezoids and summing their areas, generally providing a more accurate estimate than Riemann sums.

Example:

Using trapezoids instead of rectangles to estimate the area under a curve provides a more precise trapezoidal sum approximation.

Trigonometric Limits

Criticality: 3

Special limits involving trigonometric functions, particularly the fundamental limits like lim (sin x)/x = 1 and lim (cos x - 1)/x = 0 as x approaches 0.

Example:

When evaluating the limit of (sin(3x))/x as x approaches 0, you can manipulate the expression to use the special trigonometric limit (sin u)/u = 1.

u-substitution

Criticality: 3

A powerful integration technique that simplifies complex integrals by replacing a part of the integrand with a new variable, $u$, and transforming the differential $dx$ to $du$.

Example:

To solve $\int x \sqrt{x^2+1} , dx$ , one might use u-substitution by letting $u = x^2+1$ .