Glossary

Derivative

Criticality: 2

The instantaneous rate of change of a function with respect to its variable, representing the slope of the tangent line to the function's graph at a given point.

Example:

The derivative of $f(x) = x^2$ is $f'(x) = 2x$ , which tells us the slope of the tangent line at any point on the parabola.

Indeterminate Form

Criticality: 3

A mathematical expression that does not have a well-defined limit without further analysis, typically resulting from direct substitution into a limit expression.

Example:

When evaluating $\lim_{{x \to 0}} \frac{\sin(x)}{x}$ , direct substitution yields $\frac{0}{0}$ , which is an indeterminate form requiring further methods like L'Hospital's Rule.

L'Hospital's Rule

Criticality: 3

A calculus method used to evaluate limits of quotients that result in indeterminate forms like $\frac{0}{0}$ or $\frac{\pm \infty}{\pm \infty}$ by taking the derivatives of the numerator and denominator.

Example:

To find $\lim_{{x \to \infty}} \frac{e^x}{x}$ , we can apply L'Hospital's Rule by differentiating both top and bottom, getting $\lim_{{x \to \infty}} \frac{e^x}{1}$ , which clearly goes to infinity.

Limit

Criticality: 2

The value that a function approaches as the input (or index) approaches some value.

Example:

Understanding the limit of a function like $\lim_{{x \to 2}} (x^2 + 1)$ helps us see that as x gets closer to 2, the function's value gets closer to 5.

Substitution

Criticality: 1

A direct method of evaluating a limit by replacing the variable with the value it approaches.

Example:

When finding $\lim_{{x \to 3}} (x + 5)$ , we can use direct substitution to get $3 + 5 = 8$ , which is the limit.

Glossary

Derivative

Criticality: 2

The instantaneous rate of change of a function with respect to its variable, representing the slope of the tangent line to the function's graph at a given point.

Example:

The derivative of $f(x) = x^2$ is $f'(x) = 2x$ , which tells us the slope of the tangent line at any point on the parabola.

Indeterminate Form

Criticality: 3

A mathematical expression that does not have a well-defined limit without further analysis, typically resulting from direct substitution into a limit expression.

Example:

When evaluating $\lim_{{x \to 0}} \frac{\sin(x)}{x}$ , direct substitution yields $\frac{0}{0}$ , which is an indeterminate form requiring further methods like L'Hospital's Rule.

L'Hospital's Rule

Criticality: 3

Example:

Limit

Criticality: 2

The value that a function approaches as the input (or index) approaches some value.

Example:

Understanding the limit of a function like $\lim_{{x \to 2}} (x^2 + 1)$ helps us see that as x gets closer to 2, the function's value gets closer to 5.

Substitution

Criticality: 1

A direct method of evaluating a limit by replacing the variable with the value it approaches.

Example:

When finding $\lim_{{x \to 3}} (x + 5)$ , we can use direct substitution to get $3 + 5 = 8$ , which is the limit.