Problem 3 - Calculus for Biology and Medicine

Question

In Problems \(1-4\), show that each function is continuous at the given value. $$ f(x)=x^{3}+2 x+1, c=2 $$

Step-by-Step Solution

Answer

The function is continuous at \( x = 2 \) because \( \lim_{{x \to 2}} f(x) = f(2) = 13 \).

Step 1: Understand the definition of continuity

A function is continuous at a point \( c \) if the limit of \( f(x) \) as \( x \) approaches \( c \) is equal to \( f(c) \). Mathematically, \[ \lim_{{x \to c}} f(x) = f(c) \]. We need to verify this condition for the function \( f(x) = x^3 + 2x + 1 \) at \( c = 2 \).

Step 2: Calculate \( f(c) \)

Substitute \( c = 2 \) into the function to find \( f(c) \): \[ f(2) = 2^3 + 2(2) + 1 = 8 + 4 + 1 = 13 \]. Thus, \( f(2) = 13 \).

Step 3: Find the limit \( \lim_{{x \to 2}} f(x) \)

Evaluate the limit of \( f(x) \) as \( x \) approaches 2: \[ \lim_{{x \to 2}} (x^3 + 2x + 1) = (2)^3 + 2(2) + 1 = 8 + 4 + 1 = 13 \]. Therefore, \( \lim_{{x \to 2}} f(x) = 13 \).

Step 4: Verify continuity condition

Compare the calculated limit and the function value at \( c = 2 \):The limit \( \lim_{{x \to 2}} f(x) = 13 \) is equal to \( f(2) = 13 \). Since the limit exists and is equal to the function value, \( f(x) \) is continuous at \( x = 2 \).

Key Concepts

Limit of a Function

In calculus, the limit of a function is a fundamental concept that helps us understand how a function behaves as it approaches a particular point. Let's consider a function \( f(x) \) that approaches some value \( L \) as \( x \) gets closer and closer to a point \( c \). Formally, we write this as \( \lim_{{x \to c}} f(x) = L \). It's like asking, "What happens to \( f(x) \) when we get infinitely close to \( c \), but maybe not at \( c \) itself?"

There are a few important parts to remember about evaluating limits:

• If the limit exists, \( f(x) \) approaches a specific value.
• We can find limits by direct substitution, simplify using algebraic expressions, or use limit theorems.
• For polynomial functions, like in this example, we can often directly substitute the value.

By understanding limits, we are better equipped to determine continuity and the behavior of a function near a point.

Polynomial Functions

Polynomial functions are expressions that involve variables raised to positive integer powers. They have the general form \( f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0 \), where the powers are whole numbers, and the coefficients \( a_n, a_{n-1}, \dots, a_1, a_0 \) are constants. These functions are important in calculus because they are simple yet powerful enough to describe a variety of behaviors.

Characteristics of polynomial functions:

• Continuity: Polynomial functions are continuous everywhere, which means no breaks, jumps or vertical asymptotes.
• Smoothness: They do not have sharp corners or cusps.
• Easy to differentiate and integrate.

In our example, the function \( f(x) = x^3 + 2x + 1 \) is a polynomial. By plugging \( x = 2 \) into it, we find that the function value is \( 13 \), which we use to determine the function's continuity.

Definition of Continuity

The definition of continuity is crucial when analyzing the behavior of functions at specific points. A function \( f(x) \) is said to be continuous at a point \( c \) if three conditions are met:

• The function \( f(c) \) is defined, meaning \( c \) is in the domain of \( f \).
• \( \lim_{{x \to c}} f(x) \) exists, meaning the limit as \( x \) approaches \( c \) is a real number.
• The limit value is equal to the function value, \( \lim_{{x \to c}} f(x) = f(c) \).

Continuity can be thought of as being able to "draw" the function at \( x = c \) without lifting your pen. For the function \( f(x) = x^3 + 2x + 1 \), we've shown that it is continuous at \( x = 2 \) because both the limit and function value are \( 13 \). This seamless behavior of polynomials at every point is what quintessentially makes them an ideal choice for continuous functions.