Glossary

A

Average Rate of Change

Criticality: 3

The total change in the value of a function over an interval divided by the length of that interval, representing the overall trend of change.

Example:

If a company's profit increased from $1000 to$ 5000 over 4 months, the average rate of change in profit was ( $5000 -$ 1000) / 4 = $1000 per month.

C

Candidates Test

Criticality: 3

A systematic method used to determine the global maximum and minimum values of a continuous function over a closed interval.

Example:

To find the highest and lowest temperatures a specific chemical reaction reaches between 0 and 10 minutes, you'd use the candidates test on its temperature function.

Candidates test

Criticality: 2

A method used to find the absolute maximum or minimum values of a continuous function on a closed interval by evaluating the function at its critical points and endpoints.

Example:

To find the global maximum of $f(x) = x^3 - 3x$ on $[-2, 2]$ , you'd use the candidates test by checking $f(-2)$ , $f(2)$ , and $f$ at any critical points within the interval.

Closed interval

Criticality: 2

An interval that includes its endpoints, typically denoted with square brackets, e.g., [a, b].

Example:

The set of all real numbers from 0 to 10, including 0 and 10, forms a closed interval [0, 10].

Concave Down

Criticality: 3

A curve is concave down over an interval if its second derivative is less than or equal to zero (f''(x) ≤ 0) throughout that interval. Graphically, it resembles an inverted 'U' shape or a 'sad face'.

Example:

The path of a ball thrown into the air, neglecting air resistance, is concave down, forming a parabolic arc that opens downwards.

Concave Up

Criticality: 3

A curve is concave up over an interval if its second derivative is greater than or equal to zero (f''(x) ≥ 0) throughout that interval. Graphically, it resembles a 'U' shape or a 'smiley face'.

Example:

The graph of a simple quadratic function like f(x) = x² is concave up across its entire domain, always opening upwards.

Concave down

Criticality: 3

Describes a portion of a function's graph that resembles an "n" shape, where the curve opens downwards, indicated by a negative second derivative.

Example:

The graph of $f(x) = -x^2$ is concave down everywhere, as it forms an inverted bowl shape opening downwards.

Concave up

Criticality: 3

Describes a portion of a function's graph that resembles a "U" shape, where the curve opens upwards, indicated by a positive second derivative.

Example:

The graph of $f(x) = x^2$ is concave up everywhere, as it forms a bowl shape opening upwards.

Concavity

Criticality: 3

Concavity describes the direction in which a curve bends, indicating whether it opens upwards or downwards. It is fundamentally linked to the sign of the second derivative of a function.

Example:

When analyzing the path of a projectile, understanding its concavity helps determine if it's curving upwards or downwards at different points in its trajectory.

Constraints

Criticality: 3

Conditions or restrictions that must be satisfied in an optimization problem, often used to reduce the number of variables.

Example:

If you're building a fence, the total length of fencing available acts as a Constraint on the perimeter of the enclosed area.

Continuous Function

Criticality: 2

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

Example:

The function f(x) = x² is a continuous function because its graph is a smooth parabola with no interruptions.

Continuous Function

Criticality: 2

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

Example:

The path of a smoothly flying drone is represented by a continuous function because its position changes without sudden, instantaneous jumps.

Continuous function

Criticality: 3

A function whose graph can be drawn without lifting the pen, meaning it has no breaks, jumps, or holes.

Example:

The path of a smoothly flying drone from takeoff to landing represents a continuous function of its altitude over time.

Critical Point

Criticality: 3

A point on the graph of a function where the first derivative is either zero or does not exist, provided the function itself is defined at that point.

Example:

To find where a function might have a peak or a valley, you first need to identify its critical points by setting the derivative to zero.

Critical Point

Criticality: 3

A point on a graph where the first derivative of a function is zero or undefined, indicating a potential local maximum, minimum, or saddle point.

Example:

When you set the derivative of a profit function to zero, you find the Critical Point that could correspond to maximum profit.

Critical Point

Criticality: 3

A critical point of a function is where its first derivative is zero or undefined, often indicating a potential local maximum, minimum, or point of inflection.

Example:

For $f(x) = x^3 - 3x$ , the points where $f'(x) = 3x^2 - 3 = 0$ (i.e., $x = \pm 1$ ) are its critical points.

Critical Point

Criticality: 3

A point in the domain of a function where its first derivative is either zero or undefined, indicating a potential local extremum or point of inflection.

Example:

When optimizing the dimensions of a container to maximize volume, you would first find the critical points of the volume function.

Critical Points

Criticality: 3

Points in the domain of a function where its first derivative is either zero or undefined.

Example:

To find the critical points of $f(x) = x^3 - 3x$ , we set its derivative $x = \pm 1$ to zero, yielding $x = \pm 1$ .

Critical Points

Criticality: 3

Points in the domain of a function where its first derivative is either zero or undefined, often corresponding to local maxima, minima, or points of inflection.

Example:

For $f(x) = x^3 - 3x$ , the critical points are found by setting $f'(x) = 3x^2 - 3 = 0$ , which gives $x = \pm 1$ .

Critical Points

Criticality: 3

Points within a function's domain where its first derivative is either zero or undefined. These are potential locations for local and global extrema.

Example:

When optimizing the dimensions of a box to maximize volume, you'd find the critical points of the volume function to identify potential optimal sizes.

D

Decreasing (behavior)

Criticality: 3

Describes a function's behavior where its output values (y) are getting smaller as its input values (x) increase, indicated by a negative first derivative.

Example:

The function $f(x) = -x^3$ is always decreasing, as its graph continuously falls from left to right.

Decreasing Function

Criticality: 3

A function is decreasing over an interval if its output values ($f(x)$) fall as its input values ($x$) increase, meaning its first derivative is negative ($f'(x) \leq 0$).

Example:

The function $f(x) = -x^3$ is a decreasing function for all real $x$ , as its graph consistently slopes downwards.

Decreasing Intervals

Criticality: 3

These are the specific ranges of $x$-values where a function's first derivative is non-positive ($f'(x) \leq 0$), indicating the function is falling.

Example:

For $f(x) = x^2 - 4x$ , the decreasing intervals are $(-\infty, 2]$ , where the parabola slopes downwards towards its vertex.

Derivative

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.

Example:

If a function describes your distance traveled over time, its derivative would tell you your instantaneous speed at any given moment.

Derivatives

Criticality: 3

The result of differentiation, representing the instantaneous rate of change of a function at any given point.

Example:

Setting the first Derivatives of a function to zero helps locate potential maximum or minimum points.

Derivatives

Criticality: 3

Mathematical tools that represent the instantaneous rate of change of a function with respect to its independent variable, indicating the slope of the tangent line at any point.

Example:

Using derivatives, we can find the velocity of an object if its position is described by a function of time.

Differentiable

Criticality: 3

A function is differentiable at a point if its derivative exists at that point, implying the graph has a well-defined tangent line and no sharp corners or vertical tangents.

Example:

The smooth curve of a parabola is differentiable everywhere, allowing us to find the slope at any point.

Differentiable Function

Criticality: 2

A function that has a well-defined derivative at every point in its domain, implying its graph is smooth and has no sharp corners or vertical tangents.

Example:

The function f(x) = sin(x) is a differentiable function everywhere, as its slope can be found at any point.

Differentiation

Criticality: 3

A fundamental calculus operation used to find the rate at which a function's value changes with respect to its input, crucial for finding extreme values.

Example:

To find the peak height of a projectile, you would use Differentiation to determine when its vertical velocity is zero.

Domain

Criticality: 3

The complete set of all possible input values (x-values) for which a function is mathematically defined and produces a real output.

Example:

For $f(x) = \sqrt{x-3}$ , the domain is $x \ge 3$ , because the expression under the square root cannot be negative.

Downward Slope

Criticality: 2

A downward slope on a function's graph indicates that as you move from left to right, the $y$-values are decreasing.

Example:

The graph of $f(x) = -2x + 5$ always exhibits a downward slope, showing a constant decrease in $y$ as $x$ increases.

E

Endpoints

Criticality: 2

The specific values that define the boundaries of an interval, often denoted as 'a' and 'b' in [a, b].

Example:

When analyzing a car's speed between 0 and 60 seconds, 0 and 60 are the endpoints of the time interval.

Endpoints

Criticality: 3

The boundary values of a closed interval over which a function is being analyzed. These points must be evaluated when searching for global extrema.

Example:

If you're analyzing a car's fuel efficiency between 0 and 100 km/h, 0 km/h and 100 km/h are the endpoints of your speed interval.

Even function

Criticality: 2

A function $f(x)$ is even if $f(-x) = f(x)$ for all $x$ in its domain, resulting in a graph that is symmetric about the y-axis.

Example:

The function $f(x) = \cos(x)$ is an even function because $\cos(-x) = \cos(x)$ , and its graph is symmetric about the y-axis.

Extrema

Criticality: 2

Refers to the maximum and minimum points on the graph of a function.

Example:

When analyzing the trajectory of a projectile, the highest point it reaches is an extrema.

Extreme Value Theorem

Criticality: 3

A fundamental theorem in calculus stating that a continuous function on a closed interval must attain both a maximum and a minimum value within that interval.

Example:

If you track the temperature of a specific room over a 24-hour period, the Extreme Value Theorem guarantees there was a highest and lowest temperature recorded during that time.

Extreme Value Theorem

Criticality: 3

A fundamental theorem in calculus that guarantees a continuous function on a closed interval will attain both a global maximum and a global minimum.

Example:

Because the height of a roller coaster is a continuous function over its track length, the Extreme Value Theorem ensures there's a highest and lowest point on the ride.

F

First Derivative

Criticality: 3

The first derivative of a function, $f'(x)$, describes the instantaneous rate of change of $f(x)$ with respect to $x$.

Example:

To find the velocity of a car given its position function, you would calculate the first derivative of the position function.

First Derivative

Criticality: 3

The first derivative of a function represents its instantaneous rate of change at any given point, indicating the slope of the tangent line and whether the function is increasing or decreasing.

Example:

If the first derivative of a car's position function is positive, it means the car is currently moving forward.

First Derivative Test

Criticality: 3

A calculus method used to classify critical points of a function as local maxima, local minima, or points of inflection by analyzing the sign changes of the first derivative.

Example:

Applying the First Derivative Test to $f(x) = x^3 - 3x$ , we can determine that $x=-1$ corresponds to a local maximum and $x=1$ to a local minimum.

First Derivative Test

Criticality: 3

The First Derivative Test is a method used to determine the local maxima and minima of a function by analyzing the sign changes of its first derivative around critical points.

Example:

Using the First Derivative Test, you can confirm that if $f'(x)$ changes from positive to negative at a critical point, it indicates a local maximum.

First Derivative Test

Criticality: 2

A method used to classify critical points by examining the sign change of the first derivative around the critical point, especially when the Second Derivative Test is inconclusive.

Example:

If the second derivative test yields zero for a critical point, you would then use the first derivative test to determine if it's a minimum, maximum, or point of inflection.

G

General shapes

Criticality: 2

Refers to the characteristic visual form of different types of function graphs, such as parabolas for quadratics or S-curves for cubics.

Example:

Understanding the general shape of a cubic function helps predict its behavior, like having at most two turning points.

Global (Absolute) Extrema

Criticality: 3

The highest (maximum) or lowest (minimum) value of the function over its entire domain.

Example:

For the function f(x) = x^2 over all real numbers, the point (0,0) is the global minimum.

Global Extrema

Criticality: 3

The absolute largest (global maximum) and smallest (global minimum) values that a function attains over a specified interval.

Example:

For a projectile's height over time, the global extrema would be its maximum height and the height when it lands (or starts, if that's lower).

Global Maximum

Criticality: 3

The single highest value that a function achieves over a given closed interval.

Example:

In a profit function for a company, the global maximum would represent the highest possible profit achievable within a specific production range.

Global Minimum

Criticality: 3

The single lowest value that a function achieves over a given closed interval.

Example:

For a cost function, the global minimum indicates the lowest possible cost to produce a certain item within a defined production quantity.

Graphical Analysis

Criticality: 2

Graphical analysis involves interpreting the behavior of a function and its derivatives by examining their graphs, such as identifying increasing/decreasing intervals from slopes or derivative signs.

Example:

Through graphical analysis of $f'(x)$ , one can quickly see where $f(x)$ is increasing (where $f'(x)$ is above the x-axis).

H

Horizontal asymptotes

Criticality: 3

Horizontal lines that a function's graph approaches as the input variable (x) tends towards positive or negative infinity.

Example:

The function $y = \frac{2x^2+1}{x^2-4}$ has a horizontal asymptote at $y=2$ , indicating the graph flattens out at this height for very large or very small x-values.

I

Increasing (behavior)

Criticality: 3

Describes a function's behavior where its output values (y) are getting larger as its input values (x) increase, indicated by a positive first derivative.

Example:

The function $f(x) = e^x$ is always increasing, as its graph continuously rises from left to right.

Increasing Function

Criticality: 3

A function is increasing over an interval if its output values ($f(x)$) rise as its input values ($x$) increase, meaning its first derivative is positive ($f'(x) \geq 0$).

Example:

The function $f(x) = x^2$ is an increasing function for $x \geq 0$ , as its graph slopes upwards.

Increasing Intervals

Criticality: 3

These are the specific ranges of $x$-values where a function's first derivative is non-negative ($f'(x) \geq 0$), indicating the function is rising.

Example:

For $f(x) = x^2 - 4x$ , the increasing intervals are $[2, \infty)$ , where the parabola turns upwards.

Instantaneous Rate of Change

Criticality: 3

The rate at which a function's value is changing at a specific, single point, represented by the derivative of the function at that point.

Example:

If a rocket's height is given by h(t), then h'(5) would represent the rocket's instantaneous rate of change (velocity) exactly 5 seconds after launch.

Interpret the Answer

Criticality: 2

The final step in solving an optimization problem, where the mathematical solution is translated back into the context of the original real-world scenario.

Example:

After finding 'r=10' as the radius that minimizes perimeter, you must Interpret the Answer by stating that 'the minimum perimeter is 20π meters when the radius is 10 meters'.

L

Limits

Criticality: 3

The value that a function approaches as its input approaches some value, or as the input approaches positive or negative infinity.

Example:

The limit of $\frac{\sin(x)}{x}$ as $x$ approaches 0 is 1, even though the function is undefined at $x=0$ .

Local (Relative) Extrema

Criticality: 3

The highest or lowest value of the function within a specific interval of the domain.

Example:

A roller coaster track might have several small hills and valleys, each representing a local extremum, even if they aren't the highest or lowest points of the entire ride.

Local Extrema

Criticality: 3

Points on a function's graph where it reaches a local maximum or minimum value within a specific interval.

Example:

For the function $f(x) = x^2$ , the point $(0,0)$ is a local extremum because it represents the lowest value in its immediate neighborhood.

Local Maximum

Criticality: 3

A point where the function's value is the highest within a specific interval, and its first derivative is typically zero or undefined.

Example:

Determining the local maximum of a profit function helps a business find the optimal price point to maximize earnings within a certain market segment.

Local Maximum

Criticality: 3

A point on a function's graph where the function's value is greater than or equal to the values at all nearby points, identified when the first derivative changes from positive to negative.

Example:

In the function $f(x) = -x^2 + 4x$ , the point $(2,4)$ is a local maximum because the derivative changes from positive to negative at $x=2$ .

Local Maximum

Criticality: 3

A point on the graph of a function where the function's value is greater than or equal to the values at all nearby points, often occurring where the first derivative is zero and the second derivative is negative.

Example:

For $f(x) = -x^2$ , the point $(0,0)$ is a local maximum because the function's value is highest there compared to its immediate surroundings.

Local Maximum

Criticality: 2

A local maximum is a point on a function's graph where the function's value is larger than at nearby points, often occurring where the first derivative changes from positive to negative.

Example:

The function $f(x) = -x^2 + 6x - 5$ has a local maximum at $x=3$ , representing the highest point in its local region.

Local Maximum

Criticality: 3

A point on a function's graph where the function's value is higher than at all nearby points, representing a 'peak' in the graph.

Example:

The highest point a projectile reaches before falling is a local maximum of its trajectory.

Local Minimum

Criticality: 3

A point where the function's value is the lowest within a specific interval, and its first derivative is typically zero or undefined.

Example:

In a cost function, finding the local minimum can help identify the production level that minimizes costs for a certain output range.

Local Minimum

Criticality: 3

A point on a function's graph where the function's value is less than or equal to the values at all nearby points, identified when the first derivative changes from negative to positive.

Example:

For $f(x) = x^2 - 6x + 5$ , the point $(3,-4)$ is a local minimum as the derivative changes from negative to positive at $x=3$ .

Local Minimum

Criticality: 3

A point on the graph of a function where the function's value is less than or equal to the values at all nearby points, often occurring where the first derivative is zero and the second derivative is positive.

Example:

For $f(x) = x^2$ , the point $(0,0)$ is a local minimum because the function's value is lowest there compared to its immediate surroundings.

Local Minimum

Criticality: 2

A local minimum is a point on a function's graph where the function's value is smaller than at nearby points, often occurring where the first derivative changes from negative to positive.

Example:

The parabola $f(x) = x^2 - 4x + 3$ has a local minimum at $x=2$ , which is the lowest point in its immediate vicinity.

Local Minimum

Criticality: 3

A point on a function's graph where the function's value is lower than at all nearby points, representing a 'valley' in the graph.

Example:

The lowest point a ball reaches when bouncing is a local minimum of its height function over time.

Local maximum

Criticality: 2

A point where a function's value is the largest within some small neighborhood around that point, but not necessarily the absolute highest over the entire domain.

Example:

The peak of a small hill on a hiking trail could be considered a local maximum in altitude.

Local minimum

Criticality: 2

A point where a function's value is the smallest within some small neighborhood around that point, but not necessarily the absolute lowest over the entire domain.

Example:

On a mountain range, a valley might represent a local minimum in elevation, even if there's a deeper valley elsewhere.

M

Maximizing

Criticality: 3

The process of finding the largest possible value of a quantity or function.

Example:

Maximizing the area of a rectangle with a fixed perimeter helps in efficient space utilization.

Maximum

Criticality: 3

The highest value a function attains within a given interval or its entire domain.

Example:

Determining the maximum profit a company can achieve by adjusting production levels involves finding the peak of a profit function.

Maximum value

Criticality: 2

The highest output value a function attains over a specified domain or interval.

Example:

In a profit function for a company, the maximum value indicates the highest possible profit achievable under given conditions.

Mean Value Theorem

Criticality: 3

A fundamental theorem in calculus that states for a continuous and differentiable function over a closed interval, there exists at least one point within that interval where the instantaneous rate of change equals the average rate of change over the entire interval.

Example:

If a car travels 100 km in 2 hours, its average rate of change (speed) is 50 km/h. The Mean Value Theorem guarantees that at some point during the trip, the car's instantaneous rate of change (speedometer reading) was exactly 50 km/h.

Minimizing

Criticality: 3

The process of finding the smallest possible value of a quantity or function.

Example:

Minimizing the cost of raw materials for a product is a key goal in manufacturing efficiency.

Minimum

Criticality: 3

The lowest value a function attains within a given interval or its entire domain.

Example:

Finding the dimensions of a box that result in the minimum surface area for a fixed volume is a common optimization task.

Minimum value

Criticality: 2

The lowest output value a function attains over a specified domain or interval.

Example:

For a roller coaster's height function, the minimum value would be the lowest point the coaster reaches on its track.

N

Non-Existent Derivative Point

Criticality: 2

A type of critical point where the first derivative of the function does not exist, often due to a sharp corner, cusp, or vertical tangent, but the function itself is defined.

Example:

The function f(x) = |x| has a non-existent derivative point at x=0 because of the sharp corner, making it a critical point.

O

Odd function

Criticality: 2

A function $f(x)$ is odd if $f(-x) = -f(x)$ for all $x$ in its domain, resulting in a graph that is symmetric about the origin.

Example:

The function $f(x) = \sin(x)$ is an odd function because $\sin(-x) = -\sin(x)$ , and its graph has rotational symmetry about the origin.

Open interval

Criticality: 2

An interval that does not include its endpoints, typically denoted with parentheses, e.g., (a, b).

Example:

All real numbers strictly between 0 and 10, excluding 0 and 10, constitute an open interval (0, 10).

Optimization Problem

Criticality: 3

A problem that involves finding the maximum or minimum value of a function, often subject to specific conditions.

Example:

Designing a cylindrical can to hold 1 liter of oil while minimizing the cost of the metal used is an example of an Optimization Problem.

P

Periodic function

Criticality: 2

A function whose values repeat in a regular pattern over its domain, defined by the condition $f(x+P) = f(x)$ for a fixed non-zero constant $P$ called the period.

Example:

The function $f(x) = \tan(x)$ is a periodic function with a period of $\pi$ , meaning its graph repeats every $\pi$ units along the x-axis.

Periodicity

Criticality: 2

The property of a function where its graph repeats itself at regular intervals, meaning $f(x+a) = f(x)$ for some constant $a$.

Example:

Trigonometric functions like sine and cosine exhibit periodicity, as their wave patterns repeat every $2\pi$ radians.

Point of Inflection

Criticality: 3

A point of inflection is a specific point on a curve where its concavity changes, transitioning from concave up to concave down or vice versa. At this point, the second derivative is typically zero or undefined, and the concavity must actually change.

Example:

On a rollercoaster track, a point of inflection might occur where the track smoothly transitions from curving upwards into a loop to curving downwards out of it.

Point of Inflection

Criticality: 2

A point on the graph of a function where the concavity changes, meaning the curve switches from bending upwards to bending downwards, or vice versa.

Example:

On a graph showing the rate of spread of a disease, the point of inflection often indicates when the rate of new infections starts to slow down.

Point of Inflection

Criticality: 2

A point on a curve where the concavity changes (from concave up to concave down or vice versa), and the first derivative does not change sign around a critical point.

Example:

For the function $f(x) = x^3$ , the origin $(0,0)$ is a point of inflection because the curve changes concavity there, even though its derivative is zero.

Point of Inflection

Criticality: 2

A point of inflection is where the concavity of a function changes, often occurring where the second derivative is zero or undefined, and sometimes where the first derivative is zero but not a local extremum.

Example:

For $f(x) = x^3$ , the origin $(0,0)$ is a point of inflection because the graph changes from concave down to concave up there.

Point of Inflection

Criticality: 3

A point on a function's graph where the concavity changes, meaning the curve switches from opening upwards to downwards, or vice versa.

Example:

On a graph showing the spread of a disease, a point of inflection might indicate when the rate of new infections starts to slow down after accelerating.

Point of Inflection (Critical)

Criticality: 2

A point where the second derivative is zero and changes sign, indicating a change in the concavity of the function, and where the first derivative is also zero.

Example:

For $f(x) = x^3$ , the origin $(0,0)$ is a critical point of inflection because $f'(0)=0$ and $f''(0)=0$ , with concavity changing from down to up.

Point of Inflection (Non-Critical)

Criticality: 2

A point where the second derivative is zero and changes sign, indicating a change in the concavity of the function, but where the first derivative is non-zero.

Example:

For $f(x) = x^3 + x$ , the origin $(0,0)$ is a non-critical point of inflection because $f''(0)=0$ and changes sign, but $f'(0)=1 \ne 0$ .

Points of Undefined Values

Criticality: 2

Specific input values for which a function's expression results in an undefined mathematical operation, such as division by zero or the square root of a negative number.

Example:

For $f(x) = \frac{1}{x-5}$ , $x=5$ is a point of undefined value because it leads to division by zero.

R

Range

Criticality: 3

The complete set of all possible output values (y-values) that a function can produce for its given domain.

Example:

For $f(x) = x^2$ , the range is $y \ge 0$ , as squaring any real number always results in a non-negative value.

Rate of Change

Criticality: 2

Describes how one quantity changes in relation to another, often represented by the derivative of a function.

Example:

The speed of a car is the Rate of Change of its distance over time.

Rate of Change

Criticality: 2

The rate of change indicates how one quantity changes in relation to another, often represented by the slope of a function.

Example:

If a company's profit function has a positive rate of change, it means their profits are increasing over time.

Rolle's Theorem

Criticality: 2

A special case of the Mean Value Theorem, stating that if a function is continuous on a closed interval, differentiable on the open interval, and has equal function values at the endpoints, then there exists at least one point within the interval where its derivative is zero.

Example:

If a ball is thrown straight up and returns to its initial height, Rolle's Theorem guarantees that at some point during its flight, its instantaneous rate of change (vertical velocity) was exactly zero, indicating it reached its peak.

S

Second Derivative

Criticality: 3

The second derivative of a function, denoted as f''(x), is the derivative of its first derivative. It is used to determine the concavity of a function and locate potential points of inflection.

Example:

To find where the rate of change of a car's velocity (its acceleration) is itself changing, you would calculate the second derivative of its position function.

Second Derivative

Criticality: 3

The derivative of the first derivative of a function, used to determine the concavity of a function and the nature of its critical points.

Example:

A positive Second Derivative at a critical point confirms that the point is a local minimum.

Second Derivative

Criticality: 3

The second derivative of a function represents the rate of change of its first derivative, indicating the concavity of the function's graph (whether it opens upwards or downwards).

Example:

A positive second derivative for a roller coaster track means it's curving upwards, like a valley, which is important for ride safety.

Second Derivative Test

Criticality: 3

A method used to classify critical points as local maxima, minima, or saddle points by evaluating the sign of the second derivative at those points.

Example:

Applying the Second Derivative Test to a function's critical point helps confirm whether it's a peak or a valley on the graph.

Second Derivative Test

Criticality: 3

A method used to classify critical points as local minima or maxima by evaluating the sign of the second derivative at those points.

Example:

To quickly determine if a function's turning point is a peak or a valley without checking surrounding values, you can apply the second derivative test.

Single Variable

Criticality: 3

In optimization, rewriting the quantity to be optimized in terms of only one independent variable, simplifying the differentiation process.

Example:

Using the given area Constraint to express the perimeter of a flower bed solely in terms of its radius allows it to be a function of a Single Variable.

Surface Area

Criticality: 2

The total area of the exterior surfaces of a three-dimensional object, frequently a quantity to be optimized.

Example:

To minimize the material cost for a box, one might need to find the dimensions that yield the smallest Surface Area for a given volume.

Symmetry

Criticality: 2

A property of a function's graph where it remains unchanged after certain transformations, such as reflection across an axis or rotation about the origin.

Example:

A parabola $y=x^2$ exhibits symmetry about the y-axis, meaning its left side is a mirror image of its right side.

U

Upward Slope

Criticality: 2

An upward slope on a function's graph indicates that as you move from left to right, the $y$-values are increasing.

Example:

When sketching the graph of $f(x) = e^x$ , you'll notice a continuous upward slope across its entire domain.

V

Vertical asymptotes

Criticality: 3

Vertical lines that a function's graph approaches infinitely closely but never touches, typically occurring where the function is undefined due to division by zero.

Example:

The function $y = \frac{1}{x}$ has a vertical asymptote at $x=0$ , as the graph gets arbitrarily close to the y-axis but never crosses it.

Volume

Criticality: 2

The amount of three-dimensional space occupied by an object, often a quantity to be optimized in geometric problems.

Example:

Calculating the Volume of a cylindrical tank is essential when designing it to hold a specific amount of liquid.