Limits, Continuity & Differentiability
Differentiability of piecewise functions
Grade 12
Question:
<p>Let \(f(x) = \begin{cases}(x+1)(x+2), & x > 0 \\ a\sin x + b\cos x, & x \leq 0\end{cases}\). If <i>f</i> is differentiable at <i>x</i> = 0, find the value of <i>a</i> − <i>b</i>.</p>
Step-by-Step Solution
Key Concept: For f to be differentiable at x = 0, it must be continuous at x = 0 AND the left and right derivatives must be equal at x = 0. Use these three conditions to find constraints on a and b.
<p><strong>Step 1: Continuity at x = 0</strong></p><p>For continuity: lim(x→0⁻) f(x) = f(0) = lim(x→0⁺) f(x)</p><p>• From right: lim(x→0⁺) (x+1)(x+2) = (1)(2) = 2</p><p>• From left: f(0) = a·sin(0) + b·cos(0) = b</p><p>Therefore: <strong>b = 2</strong></p><p><strong>Step 2: Differentiability at x = 0 (equal derivatives)</strong></p><p>For x > 0: f'(x) = (x+1) + (x+2) = 2x + 3, so f'(0⁺) = 3</p><p>For x < 0: f'(x) = a·cos(x) - b·sin(x), so f'(0⁻) = a·cos(0) - b·sin(0) = a</p><p>For differentiability: f'(0⁻) = f'(0⁺)</p><p>Therefore: <strong>a = 3</strong></p><p><strong>Step 3: Calculate a − b</strong></p><p>a − b = 3 − 2 = <strong>1</strong></p>
Correct Answer: 1