Rates of Change

Average Rate of Change

The average rate of change of a function over an interval measures how much the function's output changes per unit change in input over that interval. It is analogous to the slope of the secant line connecting two points on the graph.

• Definition: The average rate of change of a function f(x) from x = a to x = b is given by:

• Represents: The slope of the secant line between points (a, f(a)) and (b, f(b)).

• Example: If f(x) = x^2, the average rate of change from x = 1 to x = 3 is .

Instantaneous Rate of Change

The instantaneous rate of change at a point is the rate at which the function is changing at that exact point. It is the slope of the tangent line to the graph at that point.

• Definition: The instantaneous rate of change of f(x) at x = a is the derivative f'(a).

• Relationship: The instantaneous rate of change is found by taking the limit of the average rate of change as the interval shrinks to zero.

• Represents: The slope of the tangent line at (a, f(a)).

• Example: For f(x) = x^2, .

Evaluating Limits

Understanding Limits

Finding the limit of f(x) as x approaches a means determining the value that f(x) gets closer to as x gets arbitrarily close to a (from either side).

• Notation: means as x approaches a, f(x) approaches L.

• Difference from function value: does not require that f(a) = L or even that f(a) is defined.

• One-sided vs. Two-sided Limits: The two-sided limit exists if and only if both the left-hand limit and right-hand limit exist and are equal.

• Existence of Limit vs. Function Value: A limit can exist at a point even if the function is not defined there (e.g., removable discontinuity).

• When Limits Fail: Limits fail to exist if the function approaches different values from the left and right, or if it grows without bound (infinite limit), or oscillates.

Computing Limits

• First Step: Always try direct substitution first: plug x = a into f(x).

• If direct substitution fails:

  • If you get (an indeterminate form), try to factor, simplify, or use conjugates (for square roots).

  • If you get a nonzero number over zero, the limit may be infinite (vertical asymptote).

• Special Cases:

  • For limits involving square roots, multiply numerator and denominator by the conjugate.

Infinite Limits and Limits at Infinity

• Infinite Limit: Describes the behavior of f(x) as x approaches a finite value and f(x) increases or decreases without bound.

• Limit at Infinity: Describes the behavior of f(x) as x approaches infinity or negative infinity.

• Evaluating Infinite Limits: Analyze the denominator and numerator as x approaches the value causing division by zero.

• Evaluating Limits at Infinity for Polynomials: The highest degree term dominates as x becomes large.

• Limits at Infinity for Rational Functions:

  • If degrees are equal: limit is the ratio of leading coefficients.

  • If numerator degree < denominator degree: limit is 0.

  • If numerator degree > denominator degree: limit is infinite.

Working with Limits

Asymptotes

• Vertical Asymptote: Occurs at x = a if f(x) approaches infinity as x approaches a from either side. Typically found where the denominator of a rational function is zero and the numerator is nonzero.

• Horizontal Asymptote: Describes the value f(x) approaches as x goes to infinity or negative infinity. Determined by comparing degrees of numerator and denominator in rational functions.

Special Limits and Theorems

• Special Limits: Certain limits, such as , are fundamental and used frequently in calculus.

• Squeeze Theorem: If f(x) is "squeezed" between two functions that have the same limit at a point, then f(x) has that limit as well.

Continuity and Discontinuity

• Continuity at a Point: A function f(x) is continuous at x = a if:

  • f(a) is defined

  • exists

• Types of Discontinuity:

  • Removable: Limit exists, but function is not defined or not equal to the limit at that point.

  • Jump: Left and right limits exist but are not equal.

  • Infinite: Function approaches infinity at the point.

Finding the Slope of a Tangent Line Using Limits

• The slope of the tangent line to f(x) at x = a is given by the derivative:

Introduction to Derivatives

Definition and Interpretation

• Derivative: The derivative of f(x) at x = a is the instantaneous rate of change of f at a, or the slope of the tangent line at that point.

• Notation: or

• Interpretation: The derivative at a point gives the best linear approximation to the function near that point.

Finding the Derivative Using Limits

• The derivative is defined as the following limit:

• Example: For f(x) = x^2, .