{:[f^(')(x)=3x^(2)-2x+7" and "],[f(6)=200.],[f(1)=]:}

Find f(1): To find the value of f(1), we need to integrate the derivative f'(x) to get the original function f(x). Then we can use the given value f(6) = 200 to find the constant of integration.Integrate f'(x): Integrate f'(x) = 3x^2 - 2x + 7 with respect to x to find f(x). ∫(3x^2 - 2x + 7) dx = (3/3)x^3 - (2/2)x^2 + 7x + C f(x) = x^3 - x^2 + 7x + C, where C is the constant of integration.Find Constant C: Use the given value f(6) = 200 to find the constant C. f(6) = 6^3 - 6^2 + 7*6 + C = 200 216 - 36 + 42 + C = 200 222 + C = 200 C = 200 - 222 C = -22Calculate f(1): Now that we have the constant C, we can find f(1). f(1) = 1^3 - 1^2 + 7*1 - 22 f(1) = 1 - 1 + 7 - 22 f(1) = 7 - 22 f(1) = -15

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Math Problems

Calculus

Find derivatives of sine and cosine functions

Full solution

\begin{array}{l}f^{\prime}(x)=3 x^{2}-2 x+7 \text { and } f(6)=200 . \\ f(1)=\end{array}

Find $f(1)$ : To find the value of $f(1)$ , we need to integrate the derivative $f'(x)$ to get the original function $f(x)$ . Then we can use the given value $f(6) = 200$ to find the constant of integration.
Integrate $f'(x)$ : Integrate $f'(x) = 3x^2 - 2x + 7$ with respect to $x$ to find $f(x)$ .
$\int(3x^2 - 2x + 7) \, dx = \left(\frac{3}{3}\right)x^3 - \left(\frac{2}{2}\right)x^2 + 7x + C$
$f(x) = x^3 - x^2 + 7x + C$ , where $C$ is the constant of integration.
Find Constant C: Use the given value $f(6) = 200$ to find the constant C. $f(6) = 6^3 - 6^2 + 7\cdot 6 + C = 200$ $216 - 36 + 42 + C = 200$ $222 + C = 200$ $C = 200 - 222$ $C = -22$
Calculate $f(1)$ : Now that we have the constant $C$ , we can find $f(1)$ .
$f(1) = 1^3 - 1^2 + 7 \cdot 1 - 22$
$f(1) = 1 - 1 + 7 - 22$
$f(1) = 7 - 22$
$f(1) = -15$