Glossary

Basic Function Behavior

Criticality: 2

Refers to the inherent properties and range of common functions, such as trigonometric functions like sine and cosine always being between -1 and 1.

Example:

Knowing the basic function behavior of $\cos(x)$ allows us to immediately state that $-1 \leq \cos(1/x) \leq 1$ , which is crucial for setting up Squeeze Theorem problems.

Bounding functions

Criticality: 2

Two functions that 'sandwich' or 'squeeze' a third function, meaning the third function's values are always between the values of the two bounding functions over a given interval.

Example:

In the Squeeze Theorem, $-|x|$ and $|x|$ serve as bounding functions for $x\cos(1/x)$ as $x \to 0$ .

Chain Rule

Criticality: 3

A rule used to find the derivative of a composite function, stating that if $y = f(g(x))$, then $y' = f'(g(x)) \cdot g'(x)$.

Example:

When differentiating $f(x) = \cos(x^2)$ , you must use the Chain Rule because it's a function within a function.

Continuous

Criticality: 3

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

Example:

For a function to have continuity at $x=2$ , we need $\lim_{x \to 2} f(x)$ to exist, $f(2)$ to be defined, and $\lim_{x \to 2} f(x) = f(2)$ .

Limits

Criticality: 3

The value that a function approaches as the input (x) approaches a certain value. It describes the behavior of a function near a specific point.

Example:

Understanding limits helps us see that as $x$ gets closer and closer to 2, the function $f(x) = x+1$ gets closer and closer to 3.

Oscillating functions

Criticality: 2

Functions that repeatedly vary between two extremes, often seen with trigonometric functions, especially when their argument approaches infinity or a point of discontinuity.

Example:

The function $\sin(1/x)$ is an oscillating function as $x \to 0$ , making direct limit evaluation difficult and highlighting the need for the Squeeze Theorem.

Product Rule

Criticality: 3

A rule used to find the derivative of a product of two functions, stating that if $u(x)$ and $v(x)$ are differentiable functions, then $(u(x)v(x))' = u'(x)v(x) + u(x)v'(x)$.

Example:

To find the derivative of $f(x) = x^2 \sin(x)$ , you would apply the Product Rule, letting $u=x^2$ and $v=\sin(x)$ .

Squeeze Theorem

Criticality: 3

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

Example:

To find $\lim_{x \to 0} x^2 \sin(1/x)$ , we can use the Squeeze Theorem by bounding $x^2 \sin(1/x)$ between $-x^2$ and $x^2$ , both of which approach 0 as $x \to 0$ .

Twice-differentiable

Criticality: 2

A function is twice-differentiable if its first derivative exists and is also differentiable. This implies the function itself is continuous and its first derivative is also continuous.

Example:

If a function describes the position of a car, being twice-differentiable means we can find its velocity (first derivative) and its acceleration (second derivative), and both are smooth.

Glossary

Basic Function Behavior

Criticality: 2

Refers to the inherent properties and range of common functions, such as trigonometric functions like sine and cosine always being between -1 and 1.

Example:

Knowing the basic function behavior of $\cos(x)$ allows us to immediately state that $-1 \leq \cos(1/x) \leq 1$ , which is crucial for setting up Squeeze Theorem problems.

Bounding functions

Criticality: 2

Two functions that 'sandwich' or 'squeeze' a third function, meaning the third function's values are always between the values of the two bounding functions over a given interval.

Example:

In the Squeeze Theorem, $-|x|$ and $|x|$ serve as bounding functions for $x\cos(1/x)$ as $x \to 0$ .

Chain Rule

Criticality: 3

A rule used to find the derivative of a composite function, stating that if $y = f(g(x))$, then $y' = f'(g(x)) \cdot g'(x)$.

Example:

When differentiating $f(x) = \cos(x^2)$ , you must use the Chain Rule because it's a function within a function.

Continuous

Criticality: 3

Example:

For a function to have continuity at $x=2$ , we need $\lim_{x \to 2} f(x)$ to exist, $f(2)$ to be defined, and $\lim_{x \to 2} f(x) = f(2)$ .

Limits

Criticality: 3

The value that a function approaches as the input (x) approaches a certain value. It describes the behavior of a function near a specific point.

Example:

Understanding limits helps us see that as $x$ gets closer and closer to 2, the function $f(x) = x+1$ gets closer and closer to 3.

Oscillating functions

Criticality: 2

Functions that repeatedly vary between two extremes, often seen with trigonometric functions, especially when their argument approaches infinity or a point of discontinuity.

Example:

The function $\sin(1/x)$ is an oscillating function as $x \to 0$ , making direct limit evaluation difficult and highlighting the need for the Squeeze Theorem.

Product Rule

Criticality: 3

A rule used to find the derivative of a product of two functions, stating that if $u(x)$ and $v(x)$ are differentiable functions, then $(u(x)v(x))' = u'(x)v(x) + u(x)v'(x)$.

Example:

To find the derivative of $f(x) = x^2 \sin(x)$ , you would apply the Product Rule, letting $u=x^2$ and $v=\sin(x)$ .

Squeeze Theorem

Criticality: 3

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

Example:

To find $\lim_{x \to 0} x^2 \sin(1/x)$ , we can use the Squeeze Theorem by bounding $x^2 \sin(1/x)$ between $-x^2$ and $x^2$ , both of which approach 0 as $x \to 0$ .

Twice-differentiable

Criticality: 2

A function is twice-differentiable if its first derivative exists and is also differentiable. This implies the function itself is continuous and its first derivative is also continuous.

Example:

If a function describes the position of a car, being twice-differentiable means we can find its velocity (first derivative) and its acceleration (second derivative), and both are smooth.