Given {eq}f''(x) = 6x -2 {/eq} and {eq}f'(-1) = 3 {/eq} and {eq}f(-1) = -4 {/eq}. Find {eq}f'(x) {/eq} and {eq}f(3) {/eq}.


Given {eq}f''(x) = 6x -2 {/eq} and {eq}f'(-1) = 3 {/eq} and {eq}f(-1) = -4 {/eq}. Find {eq}f'(x) {/eq} and {eq}f(3) {/eq}.

Derivative's Solution:

If the second derivative function is given, then we'll get the two constant of integration C (constant for first derivative function) and D (constant for original function) respectively so we'll compute these constants using initial values.

{eq}\begin{align*} \displaystyle \int f''(x)\ dx&=f'(x)+C\\ \displaystyle \int( f'(x)+C)\ dx&=f(x)+Cx+D\\ \end{align*} {/eq}

Answer and Explanation: 1

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We are given:

The second derivative function and initial values are:

{eq}f''(x) = 6x -2\\ f'(-1) = 3\\ f(-1) = -4 {/eq}

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Indefinite Integrals as Anti Derivatives


Chapter 12 / Lesson 11

Indefinite integrals, an integral of the integrand which does not have upper or lower limits, can be used to identify individual points at specific times. Learn more about the fundamental theorem, use of antiderivatives, and indefinite integrals through examples in this lesson.

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