Applications of Derivatives
Chain Rule and Composition
Grade 12
Question:
<p>Let f(x) = ∫₋ₘ² e^((1+t)²) dt and g(x) = f(h(x)), where h(x) is defined for all x ∈ ℝ. If g'(2) = e⁴ and h'(2) = 1, then the absolute value of the sum of all possible values of h(2) is ________.</p>
Step-by-Step Solution
Key Concept: Apply the chain rule to composite functions and use the fundamental theorem of calculus to find f'(x). Match exponential expressions by equating exponents.
<p><strong>Step 1:</strong> Find f'(x) using Leibniz rule:<br/>f'(x) = e^((1+x)²)</p><p><strong>Step 2:</strong> By chain rule, g'(x) = f'(h(x)) · h'(x)<br/>g'(x) = e^((1+h(x))²) · h'(x)</p><p><strong>Step 3:</strong> At x = 2:<br/>g'(2) = e^((1+h(2))²) · h'(2) = e⁴<br/>e^((1+h(2))²) · 1 = e⁴<br/>e^((1+h(2))²) = e⁴</p><p><strong>Step 4:</strong> Equating exponents:<br/>(1 + h(2))² = 4<br/>1 + h(2) = ±2</p><p><strong>Step 5:</strong> Solving for h(2):<br/>h(2) = 1 or h(2) = -3<br/>Sum of all possible values = 1 + (-3) = -2<br/>Absolute value = <strong>2</strong></p>
Correct Answer: 2