Problem 5 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

Exercise 4.1 .5 a. Draw the graph of \(y_{1}\). b. Find a number \(A\) such that the graph of \(y_{2}\) is continuous. $$ \text { a. } \quad y_{1}(x)=\left\\{\begin{array}{rll} x^{2} & \text { for } x<2 \\ 3-x & \text { for } 2 \leq x \end{array} \quad\right. \text { b. } y_{2}(x)=\left\\{\begin{aligned} x^{2} & \text { for } x<2 \\ A-x & \text { for } 2 \leq x \end{aligned}\right. $$

Step-by-Step Solution

Answer

Graph \(y_1(x)\), a piecewise split at \(x=2\); Choose \(A = 6\) for \(y_2(x)\) continuity.

Step 1: Identify the Functions in y1

The function \(y_1(x)\) is a piecewise function with two expressions: \(x^2\) for \(x < 2\) and \(3 - x\) for \(x \geq 2\). Notice the split at \(x = 2\), which is where the function changes from quadratic to linear.

Step 2: Plot y1 for x < 2

For \(x < 2\), the function \(y_1(x) = x^2\) is a simple parabola opening upwards. Calculate several points, such as at \(x = 0\), \(y_1(0) = 0^2 = 0\) and \(x = 1\), \(y_1(1) = 1^2 = 1\), to build a portion of the parabola leading up to near \(x = 2\). Do not include \(x = 2\) as it's only valid for \(x < 2\).

Step 3: Plot y1 for x ≥ 2

For \(x \geq 2\), use the linear expression \(y_1(x) = 3 - x\). Calculate points like \(x = 2\), \(y_1(2) = 3 - 2 = 1\) and \(x = 3\), \(y_1(3) = 3 - 3 = 0\). Note that the behavior changes at \(x = 2\), so at that point, the function value should stay consistent between the segments.

Step 4: Find Continuity Condition for y2

For \(y_2(x)\) to be continuous, its value must be the same on both sides of \(x = 2\). Set \((\text{value from } x^2) = (\text{value from } A-x)\) at \(x = 2\), giving us \(2^2 = A - 2\). Solve for \(A\).

Step 5: Solve for A

Substitute \(x = 2\) into the continuity equation: from the left, \(x^2 = 2^2 = 4\); from the right, it is \(A - 2\). Setting these equal, \(4 = A - 2\), solve for \(A\) by adding \(2\) to both sides, giving \(A = 6\).

Step 6: Confirm Solution

Verify that with \(A = 6\), both parts of \(y_2\) connect at \(x = 2\): \(y_2(2 - 0.01) \approx 4\) and \(y_2(2) = 6 - 2 = 4\). Thus, \(y_2(x)\) is continuous at \(x = 2\).

Key Concepts

Piecewise Functions

In mathematics, a piecewise function is defined by multiple distinct sub-functions, each applicable to a specific interval of the function's domain. This allows for different behaviors of the function across different inputs. For example, consider the exercise provided where we have functions \(y_1(x)\) and \(y_2(x)\). Both are piecewise functions:

• For \(x < 2\), both \(y_1(x)\) and \(y_2(x)\) follow the quadratic expression \(x^2\).
• For \(x \geq 2\), \(y_1(x)\) uses the expression \(3 - x\), while \(y_2(x)\) uses \(A - x\) with the variable \(A\) that we found to ensure continuity.

The hallmark of piecewise functions is their ability to describe situations where a single expression wouldn't suffice. They are quite common in modeling real-world phenomena where conditions change based on certain factors, like temperature or speed.

Graphical Analysis

Graphical analysis of piecewise functions involves plotting the separate segments of each function to visually understand how they behave. Let’s break down the steps for our exercise:

• For \(x < 2\), the segment \(x^2\) forms part of a parabola opening upwards. To sketch this, identify key points like \((0, 0)\), \((1, 1)\), and approach but not include \(x = 2\).
• For \(x \geq 2\), the analysis shifts to the linear expression. For \(y_1(x) = 3 - x\), this part starts from \(x = 2\) since it is defined for \(2 \leq x\). Points such as \((2, 1)\) and \((3, 0)\) help in plotting the straight line.

When analyzing for continuity in \(y_2(x)\), ensure that the graph is seamless at the transition point \(x = 2\). This was achieved by finding \(A = 6\) to precisely connect the graph segments with no breaks.

Quadratic and Linear Functions

Quadratic and linear functions are two fundamental types of expressions in mathematics crucial in forming piecewise functions. Let's delve into their properties as seen in the exercise:

• **Quadratic Function:** Expressed as \(x^2\), this represents a parabola opening upwards. In the piecewise function, it controls behavior for \(x < 2\). Quadratics are known for their symmetry and vertex, acting as a smooth transition in graphs.
• **Linear Function:** Uses expressions like \(3 - x\) and \(A - x\). It is characterized by a constant slope, forming a straight line. In piecewise functions, linear segments can adjust to maintain continuity and are simple to calculate for any input \(x\).

In our task, understanding these forms was vital. The quadratic segment predicts the curve behavior for certain inputs, while the linear portion ensures the function doesn’t abruptly change, highlighted by calculating \(A\) for a seamless line drawing in \(y_2(x)\).