Calculus CIA 3 Review

A series of flashcards covering key calculus concepts introduced in the review notes.

Tangent Line Equation

The equation of a line that touches a curve at a point without crossing it.

Mean Value Theorem

If a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one c in (a, b) such that f'(c) = (f(b) - f(a))/(b - a).

Critical Points

Points where the derivative of a function is zero or undefined, indicating potential local maxima or minima.

Absolute Extrema

The highest (maximum) or lowest (minimum) values of a function on a closed interval.

Linear Approximation

A method using the tangent line at a point to estimate the value of a function near that point.

Increasing Function

A function is increasing on an interval if its derivative is positive on that interval.

Decreasing Function

A function is decreasing on an interval if its derivative is negative on that interval.

Points of Inflection

Points where the concavity of a function changes, determined by the second derivative.

Local Maxima

Points where a function value is higher than all nearby points.

Local Minima

Points where a function value is lower than all nearby points.

Second Derivative Test

A test to determine the concavity of a function and to identify local extrema by evaluating the sign of the second derivative.

Tangent Line Approximation

Using the slope of the tangent line at a given point to approximate values of the function.

Extrema on Closed Interval

Finding maximum and minimum values of a function that is confined to a specific range of values.

Concavity

The direction in which the curve of a function bends, determined by the sign of the second derivative.

Slope of the Tangent Line

The derivative of a function evaluated at a specific point gives the slope of the tangent line at that point.

Derivative

A measure of how a function changes as its input changes, representing the slope of the function's graph.

Linearization

An approximation of a function near a point using the tangent line at that point.

Extreme Value Theorem

A theorem stating that if a function is continuous on a closed interval, it must attain a maximum and minimum value.

Local Extrema

Points where a function takes on values higher or lower than those around them, also found via critical points.

Inflection Points

Points on the graph of a function where the curvature changes direction.

Absolute maximum value

The highest point over the entire domain of a function.

Absolute minimum value

The lowest point over the entire domain of a function.