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Find the function \(f(x)\) described by the given initial value problem.

\(f'(x) = \sin x, \quad f(0) = 10\)

 

\(f(x) =\) ?

 May 3, 2022
 #1
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Integrating \(f'(x)\) with respect to \(x\) yields

\(f(x) = -\cos(x) + C\),

where \(C\) is a constant we can solve for given the initial condition. Because we are given \(f(0) = 10,\) we can say

\(10 = f(0) = -\cos(0) + C = -1 + C\).

This implies that \(C = 11.\) Hence,

\(f(x) = -\cos(x) + 11.\)

 May 3, 2022

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