{:[f^(')(x)=6x^(5)+7" and "],[f(-2)=30.]:} f(1)=

Integrate f'(x): To find f(1), we need to integrate the derivative f'(x) to get the original function f(x). The derivative f'(x) is given as 6x^(5) + 7. Integrate f'(x) with respect to x to find f(x). ∫(6x^(5) + 7) dx = ∫6x^(5) dx + ∫7 dx = (6/6)x^(6) + 7x + C = x^(6) + 7x + C, where C is the constant of integration.Find Constant C: We know that f(-2) = 30. We can use this information to find the constant C. Substitute x = -2 and f(x) = 30 into the equation f(x) = x^(6) + 7x + C. 30 = (-2)^(6) + 7(-2) + C 30 = 64 - 14 + C 30 = 50 + C C = 30 - 50 C = -20Write Complete Function: Now that we have the constant C, we can write the complete function f(x). f(x) = x^(6) + 7x - 20Substitute x = 1: To find f(1), substitute x = 1 into the function f(x). f(1) = (1)^(6) + 7(1) - 20 f(1) = 1 + 7 - 20 f(1) = 8 - 20 f(1) = -12

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Math Problems

Algebra 2

Simplify variable expressions using properties

Full solution

\begin{array}{l}f^{\prime}(x)=6 x^{5}+7 \text { and } f(-2)=30 .\end{array}

\newline

f(1)=

Integrate $f'(x)$ : To find $f(1)$ , we need to integrate the derivative $f'(x)$ to get the original function $f(x)$ . The derivative $f'(x)$ is given as $6x^{5} + 7$ . Integrate $f'(x)$ with respect to $x$ to find $f(x)$ . $\int(6x^{5} + 7) dx = \int 6x^{5} dx + \int 7 dx = (\frac{6}{6})x^{6} + 7x + C = x^{6} + 7x + C$ , where $f(1)$ $0$ is the constant of integration.
Find Constant C: We know that $f(-2) = 30$ . We can use this information to find the constant $C$ . $\newline$ Substitute $x = -2$ and $f(x) = 30$ into the equation $f(x) = x^{6} + 7x + C$ . $\newline$ $30 = (-2)^{6} + 7(-2) + C$ $\newline$ $30 = 64 - 14 + C$ $\newline$ $30 = 50 + C$ $\newline$ $C = 30 - 50$ $\newline$ $C = -20$
Write Complete Function: Now that we have the constant $C$ , we can write the complete function $f(x)$ . $f(x) = x^{6} + 7x - 20$
Substitute $x = 1$ : To find $f(1)$ , substitute $x = 1$ into the function $f(x)$ .
$f(1) = (1)^{6} + 7(1) - 20$
$f(1) = 1 + 7 - 20$
$f(1) = 8 - 20$
$f(1) = -12$