Problem 1 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The function \( f(x) = 2x \) is continuous at \( x = 1 \).

Step 1: Understand the Definition of Continuity at a Point

A function \( f(x) \) is continuous at a point \( x = c \) if the following three conditions are met: (1) \( f(c) \) is defined, (2) \( \lim_{{x \to c}} f(x) \) exists, and (3) \( \lim_{{x \to c}} f(x) = f(c) \).

Step 2: Verify the Function is Defined at \( c \)

For the function \( f(x) = 2x \) at \( c = 1 \), calculate \( f(1) \). Substitute \( x = 1 \) into the function: \( f(1) = 2 \times 1 = 2 \). Thus \( f(1) \) is defined.

Step 3: Calculate the Limit \( \lim_{{x \to 1}} f(x) \)

Find the limit of \( f(x) = 2x \) as \( x \) approaches 1. Substituting \( x = 1 \) directly into the equation gives \( \lim_{{x \to 1}} 2x = 2 \times 1 = 2 \). Thus, the limit exists and is equal to 2.

Step 4: Check if the Limit Equals the Function Value at \( c \)

Compare the results from Step 2 and Step 3. We have \( f(1) = 2 \) and \( \lim_{{x \to 1}} f(x) = 2 \). Since they are equal, \( \lim_{{x \to 1}} f(x) = f(1) \).

Step 5: Conclusion of Continuity at \( c \)

Since all three conditions for continuity are satisfied: \( f(1) \) is defined, \( \lim_{{x \to 1}} f(x) \) exists, and \( \lim_{{x \to 1}} f(x) = f(1) \), the function \( f(x) = 2x \) is continuous at \( c = 1 \).

Key Concepts

Definition of Continuity

In mathematics, a function is said to be continuous at a certain point if the graph of the function does not have any breaks, jumps, or holes at that point. The formal definition of continuity helps us verify this property for a given function. A function \(f(x)\) is continuous at \(x = c\) if it satisfies three specific conditions:

• First, the function value \(f(c)\) must be defined. This means the function must have a valid output value for the input \(c\).
• Second, the limit of \(f(x)\) as \(x\) approaches \(c\) must exist. This implies that as \(x\) gets closer and closer to \(c\), \(f(x)\) should approach a single numerical value.
• Finally, the limit of the function as \(x\) approaches \(c\) should be equal to the function value at \(c\), i.e., \(\lim_{{x \to c}} f(x) = f(c)\).

When all these conditions are met, we say the function is continuous at the point \(c\). This ensures that there is no break in the graph of the function at this point.

Limits of Functions

Understanding limits is crucial to grasping the concept of continuity. A limit describes the behavior of a function as its input approaches a particular value. When we say \(\lim_{{x \to c}} f(x) = L\), it means that as \(x\) gets very close to \(c\), the values of \(f(x)\) approach the number \(L\).

• When finding a limit, sometimes we can directly substitute the point into the function if the function is well-behaved at that point. This is the case with polynomial and linear functions like \(f(x) = 2x\), where direct substitution is often possible.
• If substitution results in an undefined expression, like division by zero, other techniques such as factoring, conjugate multiplication, or L'Hopital's Rule might be needed to find the limit.

In practice, checking the existence and value of limits ensures that functions behave as expected at certain points, forming a core part of establishing continuity.

Continuity at a Point

A function being continuous at a specific point means that moving along the graph of the function and passing through that point is smooth with no interruptions. Let's break down what it means for the example provided:

• First, to verify continuity at \(x = 1\) for the function \(f(x) = 2x\), you check that \(f(1)\) is defined. Indeed, replacing \(x\) with 1 gives \(f(1) = 2\), which is a valid number.
• Next, you find the limit of the function as \(x\) approaches 1, which is \(\lim_{{x \to 1}} 2x = 2\). Here, the limit equals the actual value of the function at that point.
• Finally, because the function value at \(x = 1\) matches the limit at the same point, the function meets all criteria of continuity at that point.

Thus, the function \(f(x) = 2x\) is said to be continuous at \(x = 1\). This indicates that the graphical representation of \(f(x)\) can be drawn through the point \( (1, 2) \) without lifting the pencil off of the paper.