Problem 50 - Calculus: Graphical, Numerical, Algebraic

Question

Continuous Function Find a value for \(a\) so that the function $$f(x)=\left\\{\begin{array}{ll}{x^{2}+x+a,} & {x<1} \\ {x^{3},} & {x \geq 1}\end{array}\right.$$ is continuous.

Step-by-Step Solution

Answer

For the function to be continuous, the value of a should be -1.

Step 1: Formulate the Condition for a Continuous Function

A function \(f(x)\) is continuous at \(x=c\) if \(\lim_{x\to c^-} f(x) = f(c) = \lim_{x\to c^+} f(x)\). Here, as the function changes definitions at \(x=1\), we should apply the condition of continuity at \(x=1\). Therefore, we must ensure that \(\lim_{x\to 1^-}(x^2 + x + a) = f(1) = \lim_{x\to 1^+} x^3\).

Step 2: Evaluate the Right-hand and Left-hand Limits

In this step, replace \(x\) with 1 in the two function definitions, i.e. \(x^2 + x + a\) is the left-hand limit and \(x^3\) is the right-hand limit. The left-hand limit when \(x=1\) is \(1^2 + 1 + a = 1 + 1 + a = 2 + a\). For the right-hand limit, when \(x=1\), \(f(x) = 1^3 = 1\).

Step 3: Equate the Two Limits to Solve for \(a\)

The condition for \(f(x)\) to be continuous at \(x=1\) is that the right-hand limit equals the left-hand limit. Equating \(2 + a\) with \(1\) gives an equation for \(a\). Solving this equation gives \(a = 1 - 2 = -1\).

Key Concepts

Limits

When discussing continuous functions, the concept of limits is indispensable. Limits are used to describe the behavior of a function as it approaches a particular point. In simple terms, a limit captures what value a function is "heading towards" as the input variable gets closer and closer to a specific point.

To illustrate, consider a function approaching a particular value of \(x\). If this function nears the same value from both sides of that point, we say it has a limit at that point. For our exercise, we were focusing on the point \(x=1\), which required us to find the left-hand and right-hand limits for the piecewise function:

• The left-hand limit, \(\lim_{{x \to 1^-}}(x^2 + x + a)\), describes what happens to the function when \(x\) approaches 1 from the left.
• The right-hand limit, \(\lim_{{x \to 1^+}} x^3\), shows the function’s behavior as \(x\) nears 1 from the right.

Both limits should equate to ensure the function is continuous. This understanding of limits helps us solve for any unknowns, such as the variable \(a\) in this instance.

Continuity

Continuity is a fundamental property in calculus that describes a function that can be drawn without lifting a pen off the paper. For a function to be continuous at a point \(x = c\), three main conditions need to be satisfied:

• The function must be defined at \(x = c\).
• The limit of the function as \(x\) approaches \(c\) must exist.
• The limit of the function as \(x\) approaches \(c\) must be equal to the function's value at \(c\).

In our problem, it was crucial to ensure continuity at \(x = 1\). Here, we checked that both sides of the piecewise function approach the same value as \(x\) tends to 1. By doing this, we found that continuity relies heavily on the limits we calculated. Knowing these conditions allows us to ensure seamless function behavior at transition points in piecewise functions.

Piecewise Functions

Piecewise functions are a type of function that have different expressions for different intervals of the independent variable. They are particularly useful for modeling situations where a rule or pattern changes at certain points.

Our given function is a classic case of a piecewise function. It has two parts:

• For \(x < 1\), the function is defined as \(f(x) = x^2 + x + a\).
• For \(x \geq 1\), it's given by \(f(x) = x^3\).

This means the function behaves one way when \(x\) is less than 1, and another way when \(x\) is 1 or greater. To make these functions align perfectly at \(x = 1\), ensuring continuity, we calculated the limits at the boundary and solved for any constants such as \(a\).

Piecewise functions often reflect real-world scenarios where conditions or rules change due to certain thresholds or criteria, making them invaluable in both mathematical theory and practical applications.