Mathematics is an essential tool in pharmaceutical sciences because it helps in solving equations, analyzing data, and understanding processes like drug kinetics, enzyme reactions, and biological variations. In this unit, we explore Partial Fractions, Logarithms, Functions, and Limits & Continuity which form the foundation for advanced applications in chemical kinetics and pharmacokinetics.
Introduction to Partial Fractions
Partial fraction decomposition is a mathematical technique used to break down a complex rational fraction into a sum of simpler, or partial, fractions. This process is the reverse of adding fractions with different denominators and is particularly useful in calculus for integration and in various scientific fields for solving complex equations.
Introduction to Logarithms
A logarithm is the inverse operation to exponentiation. It’s a mathematical function that determines the power to which a base number must be raised to get another number. In simpler terms, it answers the question: “What exponent do I need to get this number?”
Definition of Logarithms
The relationship between exponents and logarithms is defined as:
If by=x, then logb(x)=y
- b is the base of the logarithm.
- y is the logarithm (the exponent).
- x is the argument of the logarithm.
Example: Because 23=8, the logarithm is log2(8)=3.
Real-Valued Function
A real-valued function is a function whose range is a subset of the set of real numbers (R). This means that for every input value from its domain, the output value is a real number. In simple terms, the function’s output can be any number on the number line.
Introduction to Limits
The concept of a limit is fundamental to calculus and analysis. It allows us to understand the behavior of a function as its input gets very close to a particular value. A limit does not care what the function’s value actually is at that point, but rather what value it is “approaching.”
For example, consider the function f(x)=x−2×2−4. This function is undefined at x=2 because it would result in division by zero. However, we can still ask: what value does f(x) get closer and closer to as x gets closer and closer to 2?
As you approach x=2 from both the left (values less than 2) and the right (values greater than 2), the function’s value gets closer to 4. Therefore, we say the limit of the function as x approaches 2 is 4.
Limit of a Function
The limit of a function f(x) as x approaches a number c is a single real number, denoted as L, if the function’s value gets arbitrarily close to L as x gets arbitrarily close to c (from either side), but not necessarily equal to c. This is written as:
limx→cf(x)=L
For the limit to exist, the function must approach the same value from both the left and the right sides of c.
- Left-hand limit: The value the function approaches as x approaches c from values less than c. It is denoted as limx→c−f(x).
- Right-hand limit: The value the function approaches as x approaches c from values greater than c. It is denoted as limx→c+f(x).
A limit exists if and only if the left-hand limit equals the right-hand limit. limx→cf(x)=L⟺limx→c−f(x)=L and limx→c+f(x)=L
Definition of a Limit of a Function
The formal, rigorous definition of a limit is known as the epsilon-delta definition. It precisely defines the concept of “arbitrarily close.”
Let f be a function defined on an open interval containing c (except possibly at c itself). The statement limx→cf(x)=L means that for every number ϵ>0 (epsilon), there exists a number δ>0 (delta) such that if the distance between x and c is less than δ (but not zero), then the distance between f(x) and L is less than ϵ.
In mathematical terms: 0<∣x−c∣<δ⟹∣f(x)−L∣<ϵ
This definition ensures that no matter how small a “target” distance (ϵ) you choose around the limit L, you can always find a corresponding “input range” (δ) around c such that all x values within that range (excluding c) produce f(x) values within your target distance.