Unit 4: Linear Approximations and Indeterminate Forms

Local Linearity

The idea that if you zoom in close enough on any differentiable curve at a specific point, the curve looks like a straight line.

Tangent Line

The straight line that represents the local approximation of a function at a given point.

Linearization

The process of finding the equation of the tangent line to estimate function values near a point.

Standard Linear Approximation

The formula used to approximate functions, denoted as L(x) = f(a) + f'(a)(x - a).

Point-Slope Form of a Line

The equation of a line given by y - y1 = m(x - x1), where m is the slope.

Function Derivative

The rate of change of a function at a particular point; used to find the slope of the tangent line.

Overestimate

When a linear approximation of a function yields a value greater than the actual value.

Underestimate

When a linear approximation of a function yields a value less than the actual value.

Concavity

The direction of the curvature of a function, influenced by the second derivative.

Second Derivative Test

A method to determine concavity, where f''(x) > 0 indicates concave up and f''(x) < 0 indicates concave down.

L'Hôpital's Rule

A method to evaluate limits that results in indeterminate forms by using derivatives.

Indeterminate Forms

Forms resulting from direct substitution that require alternative methods for evaluation, such as 0/0 or ∞/∞.

Limit of a Function

The value that a function approaches as the input approaches a given value.

Chain Rule

A rule for differentiating compositions of functions.

Independent Derivatives

The requirement to differentiate the numerator and denominator separately in L'Hôpital's Rule.

Direct Substitution

The initial step in evaluating limits by plugging in the value directly into the function.

Concave Up Function

A type of function whose graph is shaped like a cup; it lies above its tangent lines.

Concave Down Function

A type of function whose graph is shaped like a cap; it lies below its tangent lines.

Approximation Accuracy

The degree to which the linear approximation approaches the actual value of the function.

Worked Example

A detailed solution to a problem that illustrates the application of theoretical concepts.

Fractional Limits

Limits that take the form of one function divided by another; applicable in L'Hôpital's Rule.

Quotient Rule

A differentiation rule for finding the derivative of a ratio of functions, but not applicable in L'Hôpital's Rule.

Notation Error

Mistakes made in writing mathematical symbols or equations, often leading to confusion.

Exam Procedures

The required methods and formatting expectations when demonstrating calculus limits and derivatives on assessments.