Glossary

Concave Down

Criticality: 2

A section of a curve where it bends downwards, resembling an inverted cup, and its second derivative is negative.

Example:

The function $f(x) = -x^2$ is concave down everywhere, with its graph opening downwards like a frown.

Concave Up

Criticality: 2

A section of a curve where it bends upwards, resembling a cup holding water, and its second derivative is positive.

Example:

The function $f(x) = x^2$ is concave up everywhere, meaning its graph opens upwards like a smile.

Concavity

Criticality: 3

A property of a curve that describes its direction of bending, specifically whether it opens upwards or downwards.

Example:

The concavity of a function can be determined by the sign of its second derivative, indicating if the slope is increasing or decreasing.

Differentiable

Criticality: 2

A function is differentiable at a point if its derivative exists at that point, meaning it has a well-defined tangent line and no sharp corners or breaks.

Example:

The function $f(x) = |x|$ is not differentiable at $x=0$ because it has a sharp corner there, preventing a unique tangent.

Linear Approximation

Criticality: 3

Using a tangent line to estimate the value of a function at points very close to the point of tangency.

Example:

To estimate $\sqrt{4.01}$ , you can use the linear approximation of $f(x) = \sqrt{x}$ at $x=4$ , which is $y = \frac{1}{4}x + 1$ .

Local Linearity

Criticality: 3

The property of a function where its graph appears as a straight line when viewed at a sufficiently small scale around a specific point.

Example:

When you zoom in very closely on the graph of $y = \sin(x)$ near $x=0$ , it looks almost identical to the line $y=x$ , demonstrating local linearity.

Overestimate

Criticality: 2

An approximation of a value that is greater than the actual value.

Example:

If you use a tangent to approximate a function that is concave down, your approximation will typically be an overestimate of the true function value.

Second Derivative

Criticality: 2

The derivative of the first derivative of a function, used to determine the concavity of the function's graph.

Example:

If the second derivative of a function is positive over an interval, the function is concave up in that interval.

Tangent

Criticality: 3

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

Example:

The line $y = 2x - 1$ is a tangent to the parabola $y = x^2$ at the point (1, 1), indicating the instantaneous rate of change.

Underestimate

Criticality: 2

An approximation of a value that is less than the actual value.

Example:

When approximating a function that is concave up using a tangent, the result will generally be an underestimate of the actual function value.

Glossary

Concave Down

Criticality: 2

A section of a curve where it bends downwards, resembling an inverted cup, and its second derivative is negative.

Example:

The function $f(x) = -x^2$ is concave down everywhere, with its graph opening downwards like a frown.

Concave Up

Criticality: 2

A section of a curve where it bends upwards, resembling a cup holding water, and its second derivative is positive.

Example:

The function $f(x) = x^2$ is concave up everywhere, meaning its graph opens upwards like a smile.

Concavity

Criticality: 3

A property of a curve that describes its direction of bending, specifically whether it opens upwards or downwards.

Example:

The concavity of a function can be determined by the sign of its second derivative, indicating if the slope is increasing or decreasing.

Differentiable

Criticality: 2

A function is differentiable at a point if its derivative exists at that point, meaning it has a well-defined tangent line and no sharp corners or breaks.

Example:

The function $f(x) = |x|$ is not differentiable at $x=0$ because it has a sharp corner there, preventing a unique tangent.

Linear Approximation

Criticality: 3

Using a tangent line to estimate the value of a function at points very close to the point of tangency.

Example:

To estimate $\sqrt{4.01}$ , you can use the linear approximation of $f(x) = \sqrt{x}$ at $x=4$ , which is $y = \frac{1}{4}x + 1$ .

Local Linearity

Criticality: 3

The property of a function where its graph appears as a straight line when viewed at a sufficiently small scale around a specific point.

Example:

When you zoom in very closely on the graph of $y = \sin(x)$ near $x=0$ , it looks almost identical to the line $y=x$ , demonstrating local linearity.

Overestimate

Criticality: 2

An approximation of a value that is greater than the actual value.

Example:

If you use a tangent to approximate a function that is concave down, your approximation will typically be an overestimate of the true function value.

Second Derivative

Criticality: 2

The derivative of the first derivative of a function, used to determine the concavity of the function's graph.

Example:

If the second derivative of a function is positive over an interval, the function is concave up in that interval.

Tangent

Criticality: 3

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

Example:

The line $y = 2x - 1$ is a tangent to the parabola $y = x^2$ at the point (1, 1), indicating the instantaneous rate of change.

Underestimate

Criticality: 2

An approximation of a value that is less than the actual value.

Example:

When approximating a function that is concave up using a tangent, the result will generally be an underestimate of the actual function value.