Problem 67 - Calculus Early Transcendentals: Pearson New International Edition

Question

Let \(f(x)=\left\\{\begin{array}{ll}m x+b & \text { if } x<2 \\ x^{2} & \text { if } x \geq 2\end{array}\right.\) Determine \(m\) and \(b\) so that \(f\) is differentiable everywhere.

Step-by-Step Solution

Answer

The values are \(m = 4\) and \(b = -4\).

Step 1: Understand Differentiability Requirement

To ensure that the piecewise function is differentiable everywhere, it must be continuous at every point and also have equal derivatives from each side at the point where the function changes, which is at \(x = 2\).

Step 2: Ensure Continuity at x=2

For continuity at \(x = 2\), the limits from the left and right at \(x = 2\) must be equal. Thus, \(m(2) + b = 2^2\). This simplifies to \(2m + b = 4\). This is our first condition for continuity.

Step 3: Equal Derivatives from Both Sides at x=2

To ensure differentiability at \(x = 2\), the derivative from the left \((m)\) must equal the derivative from the right, which is the derivative of \(x^2\) evaluated at 2, i.e. \(2x|_{x=2} = 4\). Therefore, \(m = 4\). This is our second condition.

Step 4: Solve the System of Equations

We have two equations: \(2m + b = 4\) and \(m = 4\). Substitute \(m = 4\) into the first equation: \(2(4) + b = 4\). This simplifies to \(8 + b = 4\). Solving for \(b\), we get \(b = -4\).

Step 5: Verify Both Conditions

Check if both conditions are satisfied with the calculated values. The continuity condition \(2m + b = 4\) becomes \(8 - 4 = 4\), which is true. The differentiability condition \(m = 4\) holds as well. Thus, \(m\) and \(b\) ensure that \(f\) is differentiable everywhere.

Key Concepts

Piecewise functions

A piecewise function is a function that is defined by different expressions or formulas over different parts of its domain. They can be used to describe scenarios where a rule or behavior changes at a certain point.
For example, in the case of the function given in the exercise, we have:

• For values of \(x\) less than 2, the function is described by the linear equation \(mx + b\).
• For values of \(x\) greater than or equal to 2, the function follows the formula \(x^2\).

Piecewise functions can be a bit tricky because you have to consider each piece separately and how they connect at the points where definitions change.
Understanding these transition points is essential when analyzing properties like continuity and differentiability, as they heavily depend on how smoothly, or not, the pieces of the function join together.

Continuity

Continuity is a fundamental concept in calculus that ensures a function behaves well at a point. A function is continuous at a certain point if you can draw it without lifting your pencil from the paper.
For piecewise functions, continuity at the transition point is especially important. In our example, the function changes at \(x = 2\). For the function to be continuous at \(x = 2\), the limit from the left must equal the limit from the right.

• The left-hand limit at \(x = 2\) is determined using the first piece, \(m(2) + b\).
• The right-hand limit is simply \(2^2\).

To achieve continuity at \(x = 2\), these limits must equal each other, resulting in the equation \(2m + b = 4\).
Solving this equation alongside the conditions for differentiability will ensure that the function looks smooth and continuous as it makes the transition from one piece to the next.

Derivatives

Derivatives represent the rate at which a function is changing at any given point and are a key part of determining differentiability. For a piecewise function to be differentiable at a transition point, the derivative from both sides must be equal.
In our unique case, we evaluate the derivatives at \(x = 2\):

• The derivative from the left is determined by the linear part \(mx + b\), resulting in \(m\).
• The derivative from the right comes from the function \(x^2\), which differentiates to \(2x\) and yields 4 when \(x = 2\).

The condition for differentiability here is that these derivatives are equal, which gives us \(m = 4\).
By verifying this condition, along with ensuring continuity, we establish that the function is differentiable everywhere, meaning it is smoothly connected, and does not have any abrupt changes in slope at the transition point \(x = 2\).