Glossary

Approximation

Criticality: 2

A value or result that is close to the true value but not exact, often used when an exact calculation is difficult or impossible.

Example:

When using a finite difference method to estimate a derivative, you are finding an approximation of the true instantaneous rate of change.

Asymptote

Criticality: 2

A line that a graph approaches infinitely closely but never actually touches, indicating a point where the function is undefined or approaches infinity.

Example:

The function $f(x) = \frac{1}{x}$ has a vertical asymptote at $x=0$ and a horizontal asymptote at $y=0$ .

Average Rate of Change

Criticality: 3

The measure of how a quantity changes over a specific interval, calculated as the ratio of the change in the dependent variable to the change in the independent variable.

Example:

If a runner covers 10 kilometers in 50 minutes, their Average Rate of Change of distance is 0.2 km per minute.

Average Rate of Change

Criticality: 2

The overall rate at which a function's value changes over an interval, calculated as the slope of the secant line connecting two points on the function's graph.

Example:

If a runner covers 10 km in 1 hour, their average rate of change (speed) is 10 km/h, even if they ran faster or slower at different times.

Continuous

Criticality: 2

A function is continuous if its graph can be drawn without lifting the pen, meaning there are no breaks, holes, or jumps.

Example:

A polynomial function like $f(x) = x^2 + 3x - 5$ is continuous everywhere, as its graph is a smooth, unbroken curve.

Continuous

Criticality: 3

A function is continuous at a point if its graph can be drawn without lifting the pen, meaning the limit of the function at that point equals the function's value at that point.

Example:

A polynomial function like $f(x) = x^3 - 2x + 1$ is continuous everywhere, having no breaks or jumps.

Coordinates

Criticality: 1

A set of values that specify the exact position of a point in a plane or space, typically represented as (x, y) for 2D graphs.

Example:

The point where a graph crosses the x-axis might have coordinates like (3, 0), indicating its position.

Cusp

Criticality: 3

A point on a curve where the direction of the curve changes abruptly, creating a sharp point, and the derivative does not exist.

Example:

The graph of $f(x) = |x|$ has a sharp cusp at $x=0$ , which is why it's continuous but not differentiable there.

Derivative

Criticality: 3

The instantaneous rate of change of a function at a specific point, representing the slope of the tangent line to the function's graph at that point.

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

Criticality: 3

The derivative of a function measures its instantaneous rate of change at any point, representing the slope of the curve at that specific point.

Example:

If $f(x) = x^2$ , the derivative $f'(x) = 2x$ tells you the slope of the parabola at any x-value.

Derivative

Criticality: 3

A fundamental concept in calculus that measures the instantaneous rate at which a function's output changes with respect to its input, representing the slope of the tangent line.

Example:

The derivative of a position function with respect to time gives you the instantaneous velocity of an object.

Derivative

Criticality: 3

The derivative of a function measures the instantaneous rate at which the function's output changes with respect to its input. It is defined by a limit.

Example:

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

Differentiable

Criticality: 2

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

Example:

The absolute value function $f(x) = |x|$ is not differentiable at $x=0$ because its graph has a sharp corner there.

Differentiable

Criticality: 3

A function is differentiable at a point if its derivative exists at that point. A differentiable function has a derivative that exists at every point in its domain.

Example:

The function $f(x) = x^2$ is differentiable everywhere, meaning you can find its slope at any point.

Domain

Criticality: 2

The set of all possible input values (x-values) for which a function is defined.

Example:

For $f(x) = \sqrt{x}$ , the domain is $x \ge 0$ , as you cannot take the square root of a negative number in real numbers.

Equation of a Straight Line

Criticality: 2

The equation of a straight line is a linear relationship between x and y, commonly expressed as $y = mx + c$ (slope-intercept form) or $y - y_1 = m(x - x_1)$ (point-slope form).

Example:

The line $y = 2x + 1$ is an equation of a straight line with a slope of 2 and a y-intercept of 1.

Equation of a Tangent Line

Criticality: 3

The equation of a tangent line is a linear equation ($y = mx + c$) that describes the straight line touching a curve at a specific point, with its slope determined by the derivative at that point.

Example:

Finding the equation of a tangent line to $y = x^2$ at $(2,4)$ involves calculating the derivative at $x=2$ to get the slope, then using the point-slope form.

Function

Criticality: 2

A function is a relation where each input has exactly one output; in calculus, the derivative itself can be a new function derived from the original.

Example:

If $f(x) = x^2$ , its derivative $f'(x) = 2x$ is also a function that provides the slope for any given $x$ .

Function

Criticality: 1

A relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.

Example:

The relationship between the radius of a circle and its area, A = πr², is a function where the area is dependent on the radius.

Function Notation

Criticality: 2

A standardized way to represent functions and their inputs/outputs, typically using symbols like $f(x)$ where $f$ is the function name and $x$ is the input variable.

Example:

If a function describes the temperature over time, $T(h)$ uses Function Notation to represent the temperature at hour $h$ .

Horizontal Tangent

Criticality: 3

A horizontal tangent is a tangent line with a slope of zero, indicating a local maximum, minimum, or a saddle point on the curve.

Example:

The parabola $y = x^2$ has a horizontal tangent at its vertex $(0,0)$ , where its derivative $f'(x)=2x$ is zero.

Instantaneous Rate of Change

Criticality: 3

The instantaneous rate of change is the rate at which a function's value changes at a specific, single point, rather than over an interval.

Example:

The speedometer in a car shows your instantaneous rate of change of distance with respect to time, not your average speed over a trip.

Instantaneous Rate of Change

Criticality: 3

The rate at which a function's value is changing at a single, specific point, representing the slope of the tangent line to the function's graph at that point.

Example:

When a car's speedometer reads 60 km/h, it's showing the instantaneous rate of change of its position at that exact moment.

Limit

Criticality: 3

A value that a function approaches as the input approaches some specific value, without necessarily reaching it. It's fundamental to calculus definitions.

Example:

As you get closer and closer to a wall, the limit of your distance to the wall is zero, even if you never actually touch it.

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.

Example:

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

One-sided limits

Criticality: 2

The value a function approaches as the input approaches a point from either the left (values less than the point) or the right (values greater than the point).

Example:

For the absolute value function $f(x) = |x|$ at $x=0$ , the one-sided limit from the left is -1, and from the right is 1.

Oscillates

Criticality: 1

Describes a function whose values fluctuate rapidly between different values as the input approaches a certain point, preventing the limit from settling on a single value.

Example:

The function $f(x) = \sin(\frac{1}{x})$ oscillates infinitely many times between -1 and 1 as $x$ approaches 0, so its limit does not exist.

Slope

Criticality: 2

The measure of the steepness or incline of a line, calculated as the ratio of the vertical change (rise) to the horizontal change (run).

Example:

A road with a slope of 0.1 means that for every 10 meters horizontally, it rises 1 meter vertically.

Slope

Criticality: 3

A numerical value that describes the steepness and direction of a line, calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points.

Example:

When graphing the cost of a taxi ride versus distance, the Slope of the line represents the cost per mile.

Slope

Criticality: 3

Slope is a measure of the steepness and direction of a line, calculated as the ratio of the vertical change (rise) to the horizontal change (run).

Example:

A road sign indicating a 10% grade means the slope of the road is 0.1, rising 10 units for every 100 units horizontally.

Slope

Criticality: 2

A measure of the steepness and direction of a line, calculated as the ratio of the vertical change to the horizontal change between two points.

Example:

A ski slope with a slope of -0.5 means for every 10 meters horizontally, the elevation drops by 5 meters.

Slope of line segments

Criticality: 2

The measure of steepness between two distinct points on a graph, calculated as the change in y divided by the change in x.

Example:

When estimating the derivative of a function from a table, you might calculate the slope of line segments between adjacent data points to approximate the rate of change.

Slope of the Curve

Criticality: 3

The slope of the curve at a specific point is the steepness of the curve at that exact location, which is given by the derivative at that point.

Example:

For $f(x) = x^3$ , the slope of the curve at $x=1$ is $f'(1) = 3(1)^2 = 3$ , indicating how steeply the graph rises there.

Slope of the tangent

Criticality: 3

The steepness of the line that touches a curve at a single point, indicating the instantaneous rate of change of the function at that point.

Example:

To find the instantaneous velocity of a ball thrown upwards at 2 seconds, you'd calculate the slope of the tangent to its height-vs-time graph at $t=2$ .

Specific point

Criticality: 1

A single, precisely defined location on a graph or within a function's domain, where a property like the instantaneous rate of change is evaluated.

Example:

To find the temperature change at exactly noon, you'd look at the specific point on the temperature-time graph corresponding to 12:00 PM.

Tangent

Criticality: 3

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

Example:

Imagine a ball rolling down a hill; if it suddenly flew off, its path would be a tangent to the hill's curve at the point it left.

Twice differentiable

Criticality: 2

A function is twice differentiable if its first derivative is also continuous and differentiable, meaning the second derivative exists.

Example:

If a function describes position, being twice differentiable means not only can you find its velocity (first derivative), but also its acceleration (second derivative).

Unbounded

Criticality: 2

Describes a function whose values grow infinitely large (positive or negative) as the input approaches a certain point, causing the limit not to exist.

Example:

The function $f(x) = \frac{1}{x^2}$ is unbounded as $x$ approaches 0, as its values shoot up to infinity.

Vertical Asymptote

Criticality: 1

A vertical asymptote is a vertical line that a function's graph approaches infinitely closely but never touches or crosses, typically where the function itself is undefined.

Example:

The function $f(x) = 1/x$ has a vertical asymptote at $x=0$ , as the function's value approaches infinity as $x$ approaches 0.

Vertical Tangent

Criticality: 2

A vertical tangent is a tangent line with an undefined slope, occurring at points where the derivative approaches infinity or negative infinity.

Example:

The function $f(x) = x^{1/3}$ has a vertical tangent at $x=0$ , where its derivative $f'(x) = \frac{1}{3x^{2/3}}$ is undefined.

Vertical tangent

Criticality: 2

A point on a curve where the tangent line is vertical, meaning the slope is undefined and the derivative does not exist.

Example:

The function $f(x) = \sqrt[3]{x}$ has a vertical tangent at $x=0$ , where its graph becomes momentarily vertical.

Glossary

Approximation

Criticality: 2

A value or result that is close to the true value but not exact, often used when an exact calculation is difficult or impossible.

Example:

When using a finite difference method to estimate a derivative, you are finding an approximation of the true instantaneous rate of change.

Asymptote

Criticality: 2

A line that a graph approaches infinitely closely but never actually touches, indicating a point where the function is undefined or approaches infinity.

Example:

The function $f(x) = \frac{1}{x}$ has a vertical asymptote at $x=0$ and a horizontal asymptote at $y=0$ .

Average Rate of Change

Criticality: 3

The measure of how a quantity changes over a specific interval, calculated as the ratio of the change in the dependent variable to the change in the independent variable.

Example:

If a runner covers 10 kilometers in 50 minutes, their Average Rate of Change of distance is 0.2 km per minute.

Average Rate of Change

Criticality: 2

The overall rate at which a function's value changes over an interval, calculated as the slope of the secant line connecting two points on the function's graph.

Example:

If a runner covers 10 km in 1 hour, their average rate of change (speed) is 10 km/h, even if they ran faster or slower at different times.

Continuous

Criticality: 2

A function is continuous if its graph can be drawn without lifting the pen, meaning there are no breaks, holes, or jumps.

Example:

A polynomial function like $f(x) = x^2 + 3x - 5$ is continuous everywhere, as its graph is a smooth, unbroken curve.

Continuous

Criticality: 3

A function is continuous at a point if its graph can be drawn without lifting the pen, meaning the limit of the function at that point equals the function's value at that point.

Example:

A polynomial function like $f(x) = x^3 - 2x + 1$ is continuous everywhere, having no breaks or jumps.

Coordinates

Criticality: 1

A set of values that specify the exact position of a point in a plane or space, typically represented as (x, y) for 2D graphs.

Example:

The point where a graph crosses the x-axis might have coordinates like (3, 0), indicating its position.

Cusp

Criticality: 3

A point on a curve where the direction of the curve changes abruptly, creating a sharp point, and the derivative does not exist.

Example:

The graph of $f(x) = |x|$ has a sharp cusp at $x=0$ , which is why it's continuous but not differentiable there.

Derivative

Criticality: 3

The instantaneous rate of change of a function at a specific point, representing the slope of the tangent line to the function's graph at that point.

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

Criticality: 3

The derivative of a function measures its instantaneous rate of change at any point, representing the slope of the curve at that specific point.

Example:

If $f(x) = x^2$ , the derivative $f'(x) = 2x$ tells you the slope of the parabola at any x-value.

Derivative

Criticality: 3

A fundamental concept in calculus that measures the instantaneous rate at which a function's output changes with respect to its input, representing the slope of the tangent line.

Example:

The derivative of a position function with respect to time gives you the instantaneous velocity of an object.

Derivative

Criticality: 3

The derivative of a function measures the instantaneous rate at which the function's output changes with respect to its input. It is defined by a limit.

Example:

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

Differentiable

Criticality: 2

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

Example:

The absolute value function $f(x) = |x|$ is not differentiable at $x=0$ because its graph has a sharp corner there.

Differentiable

Criticality: 3

A function is differentiable at a point if its derivative exists at that point. A differentiable function has a derivative that exists at every point in its domain.

Example:

The function $f(x) = x^2$ is differentiable everywhere, meaning you can find its slope at any point.

Domain

Criticality: 2

The set of all possible input values (x-values) for which a function is defined.

Example:

For $f(x) = \sqrt{x}$ , the domain is $x \ge 0$ , as you cannot take the square root of a negative number in real numbers.

Equation of a Straight Line

Criticality: 2

The equation of a straight line is a linear relationship between x and y, commonly expressed as $y = mx + c$ (slope-intercept form) or $y - y_1 = m(x - x_1)$ (point-slope form).

Example:

The line $y = 2x + 1$ is an equation of a straight line with a slope of 2 and a y-intercept of 1.

Equation of a Tangent Line

Criticality: 3

The equation of a tangent line is a linear equation ($y = mx + c$) that describes the straight line touching a curve at a specific point, with its slope determined by the derivative at that point.

Example:

Finding the equation of a tangent line to $y = x^2$ at $(2,4)$ involves calculating the derivative at $x=2$ to get the slope, then using the point-slope form.

Function

Criticality: 2

A function is a relation where each input has exactly one output; in calculus, the derivative itself can be a new function derived from the original.

Example:

If $f(x) = x^2$ , its derivative $f'(x) = 2x$ is also a function that provides the slope for any given $x$ .

Function

Criticality: 1

A relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.

Example:

The relationship between the radius of a circle and its area, A = πr², is a function where the area is dependent on the radius.

Function Notation

Criticality: 2

A standardized way to represent functions and their inputs/outputs, typically using symbols like $f(x)$ where $f$ is the function name and $x$ is the input variable.

Example:

If a function describes the temperature over time, $T(h)$ uses Function Notation to represent the temperature at hour $h$ .

Horizontal Tangent

Criticality: 3

A horizontal tangent is a tangent line with a slope of zero, indicating a local maximum, minimum, or a saddle point on the curve.

Example:

The parabola $y = x^2$ has a horizontal tangent at its vertex $(0,0)$ , where its derivative $f'(x)=2x$ is zero.

Instantaneous Rate of Change

Criticality: 3

The instantaneous rate of change is the rate at which a function's value changes at a specific, single point, rather than over an interval.

Example:

The speedometer in a car shows your instantaneous rate of change of distance with respect to time, not your average speed over a trip.

Instantaneous Rate of Change

Criticality: 3

The rate at which a function's value is changing at a single, specific point, representing the slope of the tangent line to the function's graph at that point.

Example:

When a car's speedometer reads 60 km/h, it's showing the instantaneous rate of change of its position at that exact moment.

Limit

Criticality: 3

A value that a function approaches as the input approaches some specific value, without necessarily reaching it. It's fundamental to calculus definitions.

Example:

As you get closer and closer to a wall, the limit of your distance to the wall is zero, even if you never actually touch it.

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.

Example:

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

One-sided limits

Criticality: 2

The value a function approaches as the input approaches a point from either the left (values less than the point) or the right (values greater than the point).

Example:

For the absolute value function $f(x) = |x|$ at $x=0$ , the one-sided limit from the left is -1, and from the right is 1.

Oscillates

Criticality: 1

Describes a function whose values fluctuate rapidly between different values as the input approaches a certain point, preventing the limit from settling on a single value.

Example:

The function $f(x) = \sin(\frac{1}{x})$ oscillates infinitely many times between -1 and 1 as $x$ approaches 0, so its limit does not exist.

Slope

Criticality: 2

The measure of the steepness or incline of a line, calculated as the ratio of the vertical change (rise) to the horizontal change (run).

Example:

A road with a slope of 0.1 means that for every 10 meters horizontally, it rises 1 meter vertically.

Slope

Criticality: 3

A numerical value that describes the steepness and direction of a line, calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points.

Example:

When graphing the cost of a taxi ride versus distance, the Slope of the line represents the cost per mile.

Slope

Criticality: 3

Slope is a measure of the steepness and direction of a line, calculated as the ratio of the vertical change (rise) to the horizontal change (run).

Example:

A road sign indicating a 10% grade means the slope of the road is 0.1, rising 10 units for every 100 units horizontally.

Slope

Criticality: 2

A measure of the steepness and direction of a line, calculated as the ratio of the vertical change to the horizontal change between two points.

Example:

A ski slope with a slope of -0.5 means for every 10 meters horizontally, the elevation drops by 5 meters.

Slope of line segments

Criticality: 2

The measure of steepness between two distinct points on a graph, calculated as the change in y divided by the change in x.

Example:

When estimating the derivative of a function from a table, you might calculate the slope of line segments between adjacent data points to approximate the rate of change.

Slope of the Curve

Criticality: 3

The slope of the curve at a specific point is the steepness of the curve at that exact location, which is given by the derivative at that point.

Example:

For $f(x) = x^3$ , the slope of the curve at $x=1$ is $f'(1) = 3(1)^2 = 3$ , indicating how steeply the graph rises there.

Slope of the tangent

Criticality: 3

The steepness of the line that touches a curve at a single point, indicating the instantaneous rate of change of the function at that point.

Example:

To find the instantaneous velocity of a ball thrown upwards at 2 seconds, you'd calculate the slope of the tangent to its height-vs-time graph at $t=2$ .

Specific point

Criticality: 1

A single, precisely defined location on a graph or within a function's domain, where a property like the instantaneous rate of change is evaluated.

Example:

To find the temperature change at exactly noon, you'd look at the specific point on the temperature-time graph corresponding to 12:00 PM.

Tangent

Criticality: 3

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

Example:

Imagine a ball rolling down a hill; if it suddenly flew off, its path would be a tangent to the hill's curve at the point it left.

Twice differentiable

Criticality: 2

A function is twice differentiable if its first derivative is also continuous and differentiable, meaning the second derivative exists.

Example:

If a function describes position, being twice differentiable means not only can you find its velocity (first derivative), but also its acceleration (second derivative).

Unbounded

Criticality: 2

Describes a function whose values grow infinitely large (positive or negative) as the input approaches a certain point, causing the limit not to exist.

Example:

The function $f(x) = \frac{1}{x^2}$ is unbounded as $x$ approaches 0, as its values shoot up to infinity.

Vertical Asymptote

Criticality: 1

A vertical asymptote is a vertical line that a function's graph approaches infinitely closely but never touches or crosses, typically where the function itself is undefined.

Example:

The function $f(x) = 1/x$ has a vertical asymptote at $x=0$ , as the function's value approaches infinity as $x$ approaches 0.

Vertical Tangent

Criticality: 2

A vertical tangent is a tangent line with an undefined slope, occurring at points where the derivative approaches infinity or negative infinity.

Example:

The function $f(x) = x^{1/3}$ has a vertical tangent at $x=0$ , where its derivative $f'(x) = \frac{1}{3x^{2/3}}$ is undefined.

Vertical tangent

Criticality: 2

A point on a curve where the tangent line is vertical, meaning the slope is undefined and the derivative does not exist.

Example:

The function $f(x) = \sqrt[3]{x}$ has a vertical tangent at $x=0$ , where its graph becomes momentarily vertical.