Problem 41 - Calculus Early Transcendentals

Question

Let $$ f(x)=\left\\{\begin{array}{ll}{3 x^{2},} & {x \leq 1} \\ {a x+b,} & {x>1}\end{array}\right. $$ Find the values of \(a\) and \(b\) so that \(f\) will be differentiable at \(x=1\)

Step-by-Step Solution

Answer

The values are \( a = 6 \) and \( b = -3 \).

Step 1: Understand Differentiability at x=1

For the function to be differentiable at \( x = 1 \), it must be continuous at \( x = 1 \) and the left-hand derivative must equal the right-hand derivative at \( x = 1 \).

Step 2: Check Continuity at x=1

Ensure the function is continuous at \( x = 1 \). For continuity, \( \lim_{{x \to 1^-}} f(x) = \lim_{{x \to 1^+}} f(x) = f(1) \). Compute both limits: \( \lim_{{x \to 1^-}} 3x^2 = 3 \) and \( \lim_{{x \to 1^+}} (ax + b) = a + b \). Set the limits equal for continuity: \( 3 = a + b \).

Step 3: Calculate the Left-Hand Derivative

The left-hand derivative at \( x = 1 \) is \( (d/dx)[3x^2] \) evaluated at \( x = 1 \). Compute the derivative: \( \frac{d}{dx}(3x^2) = 6x \). Evaluate at \( x = 1 \) to get \( 6 \).

Step 4: Calculate the Right-Hand Derivative

The right-hand derivative at \( x = 1 \) is \( (d/dx)[ax + b] \) which is simply \( a \).

Step 5: Set Derivatives Equal

For differentiability, the left-hand derivative must equal the right-hand derivative at \( x=1 \). Therefore, set \( 6 = a \).

Step 6: Solve for b

Using the equation from continuity, \( 3 = a + b \), and substituting \( a = 6 \), solve for \( b \): \( 3 = 6 + b \) which simplifies to \( b = -3 \).

Step 7: Conclude the Values of a and b

The function \( f \) is differentiable at \( x = 1 \) if \( a = 6 \) and \( b = -3 \).

Key Concepts

Continuity

Continuity is a fundamental concept when determining if a function is differentiable. For a function to be continuous at a point, it must have no breaks, jumps, or holes. In simpler terms, you should be able to draw the function without lifting your pen around that point.

To confirm continuity for a piecewise function like the one provided, \[ f(x)=\left\{\begin{array}{ll}{3x^2,} & {x \leq 1} \ {ax+b,} & {x>1}\end{array}\right. \]we need to check the value of the function as it approaches the point from both sides. This involves assessing the limit from the left (\( x\to1^- \)) and from the right (\( x\to1^+ \)). For the function to be continuous at \( x = 1 \), both these limits and the actual value of the function at \( x = 1 \) should all be equal.

• Calculate \( \lim_{{x \to 1^-}} 3x^2 = 3 \)
• Calculate \( \lim_{{x \to 1^+}} (ax+b) = a+b \)
• Set the limits equal: \( 3 = a + b \)

This ensures that the function is continuous at \( x = 1 \).

Left-hand Derivative

The left-hand derivative provides insight into the behavior of a function as it approaches a certain point from the left. It's particularly important for piecewise functions where the formula changes at specific points. To find the left-hand derivative of a piecewise function like \( f(x) \) at \( x=1 \), we calculate the derivative of the formula for values of \( x \leq 1 \).

For \( f(x) = 3x^2 \):

• Differentiate to get \( \frac{d}{dx}(3x^2) = 6x \)
• Evaluate at \( x=1 \): \( 6(1) = 6 \)

This calculation gives us the slope of the tangent line to \( f(x) \) on the left side of \( x=1 \). The left-hand derivative at \( x=1 \) is thus \( 6 \).

Right-hand Derivative

The right-hand derivative helps analyze how a function behaves when approaching a certain point from the right. For a function to be differentiable at a point, the left-hand derivative must match the right-hand derivative. Let's focus on the right side of the function \( f(x) \) as it approaches \( x=1 \).

On this side, \( f(x) = ax + b \), and its derivative is straightforward:

• Derivative is \( \frac{d}{dx}(ax + b) = a \)

At \( x=1 \), the right-hand derivative is simply equal to \( a \). To ensure differentiability, the left-hand and right-hand derivatives must be equal, therefore, \( 6 = a \).

Piecewise Function

A piecewise function is defined differently over different intervals of its domain. This type of function can seem tricky at first, but by focusing on each piece separately, you can tackle it more easily. The provided function has two pieces:

• \( f(x) = 3x^2 \) for \( x \leq 1 \)
• \( f(x) = ax+b \) for \( x > 1 \)

Analyzing a piecewise function entails checking for continuity and differentiability at the transition points, which, in this exercise, is \( x=1 \).

By individually assessing each part:

• We find limits from the left and right to ensure continuity.
• We calculate derivatives on both sides to check differentiability.

Approaching the problem step-by-step makes piecewise functions easier to understand and analyze. By finding the values of \( a \) and \( b \) that satisfy these conditions, the function transitions smoothly at \( x=1 \) without any interruptions.