Functions

Domain and Range of Functions

Understanding the domain and range of a function is fundamental in calculus. The domain is the set of all possible input values (x-values) for which the function is defined, while the range is the set of all possible output values (y-values) the function can produce.

• Domain: The set of all real numbers x for which the function f(x) is defined.

• Range: The set of all real numbers y such that y = f(x) for some x in the domain.

• Example: For , the domain is all real numbers except x = 1 (since division by zero is undefined). The range is all real numbers except y = 0.

Limits and Continuity

Limits in Biomedical Applications

Limits are used to describe the behavior of functions as the input approaches a certain value. In biomedical contexts, limits can describe concentrations or rates as time approaches a specific value.

• Limit Notation: denotes the value that f(x) approaches as x approaches a.

• Continuity: A function is continuous at x = a if .

Intro to Derivatives

Rate of Change and Differentiation

The derivative of a function measures the rate at which the function's value changes as its input changes. In pharmacokinetics, derivatives are used to model the rate of change of drug concentration in the bloodstream.

• Definition: The derivative of f(x) with respect to x is .

• Notation: represents the rate of change of concentration C with respect to time t.

• Example: If , then .

Techniques of Differentiation

Common Derivative Rules

• Power Rule:

• Product Rule:

• Quotient Rule:

• Chain Rule:

Applications of Derivatives

Pharmacokinetics and Drug Elimination

Derivatives are used to model the elimination of drugs from the bloodstream and to find rates of change at specific times.

• Example: For , the rate of elimination at t = 2 hours is .

• Application: The total amount of drug eliminated over a time interval [a, b] can be found by integrating the rate function over that interval.

Antiderivatives & Indefinite Integrals

Finding Total Change

Antiderivatives and integrals are used to determine the total change in a quantity over time, such as the total amount of drug eliminated from the body.

• Indefinite Integral: gives the family of all antiderivatives of f(x).

• Definite Integral: gives the net change in f(x) from x = a to x = b.

• Example: The total drug eliminated from t = 0 to t = 3 hours is , where E(t) is the elimination rate.

Intro to Differential Equations

Modeling with Differential Equations

Differential equations describe how a quantity changes with respect to another variable, often time. They are widely used in modeling biological and pharmacological processes.

• First-Order Differential Equation: models exponential decay of drug concentration.

• Solution: The general solution is , where is the initial concentration.

• Example: Given , , the solution is .

Table: Discrete Probability Distribution

The following table represents a discrete probability distribution for the number of doses a patient needs per week:

Application: The probability that the patient needs 2 doses in a week is 0.5.

Additional Info

• Some questions involve statistics and probability, which are not core calculus topics but are often included in applied mathematics courses for life sciences.

• Pharmacokinetic models frequently use exponential and logarithmic functions, requiring knowledge of their derivatives and integrals.