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Question
If the marginal cost of manufacturing a certain item is given by C' (x) = \[\frac{dC}{dx}\] = 2 + 0.15 x. Find the total cost function C (x), given that C (0) = 100.
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Solution
We have,
\[\frac{dC}{dx} = 2 + 0 . 15x\]
\[ \Rightarrow dC = \left( 2 + 0 . 15x \right)dx\]
Integrating both sides with respect to x, we get
\[C = 2x + \frac{0 . 15}{2} x^2 + K . . . . . \left( 1 \right)\]
\[\text{ At }C\left( 0 \right) = 100,\text{ we have }\]
\[100 = 2\left( 0 \right) + \frac{0 . 15}{2} \left( 0 \right)^2 + K\]
\[ \Rightarrow K = 100\]
Putting the value of T in (1), we get
\[C = 2x + \frac{0 . 15}{2} x^2 + 100\]
\[ \Rightarrow C = 0 . 075 x^2 + 2x + 100\]
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