One Problem at a Time

Linearization and the Fundamental Theorem of Calculus

Problem 185: Find the linearization of

(1)   \begin{equation*} f(x) = 2 - \int_{2}^{x+1} \frac{9}{1+t}dt \end{equation*}

at x = 1.

Solution: The linearization at a point is given by

(2)   \begin{equation*} L(x) = f(a) + f'(a)(x-a)\end{equation*}

where a is the point x=a. Then

(3)   \begin{align*} L(x) & = f(a) + f'(a)(x-a) \\ & = \left(2 - \int_{2}^{1+1} \frac{9}{1+t}dt \right) + \frac{d}{dx} \left( 2 - \int_{2}^{x+1} \frac{9}{1+t}dt \right)(x-1) \\ & = 2 - \int_{2}^{2} \frac{9}{1+t}dt + \frac{-9}{1+(x+1)}\bigg|_{x=1} (x-1) \\ & = 2 - 0 +(-9/3)(x-1) \\ & = 2 - 3(x-1) \\ & = -3x + 5.\end{align*}

Hence , at x=1, the linearization is given by

(4)   \begin{equation*} L(x) = -3x + 5. \end{equation*}

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