f^(')(x)=-(4)/(x^(2)) and f(2)=4 f(1)=

Given derivative and point: We are given the derivative of a function f(x), which is f'(x) = -(4)/(x^2), and we are also given a point on the function, f(2) = 4. To find f(1), we need to integrate the derivative f'(x) to get f(x) and then evaluate f(x) at x = 1.Integrate the derivative: First, let's integrate the derivative f'(x) = -(4)/(x^2). The integral of -(4)/(x^2) with respect to x is 4/x + C, where C is the constant of integration. ∫f'(x) dx = ∫-(4)/(x^2) dx = 4/x + CFind constant of integration: Now we need to find the constant of integration C. We can do this by using the given point f(2) = 4. Let's plug x = 2 into the integrated function to solve for C. 4/2 + C = 4 2 + C = 4 C = 4 - 2 C = 2Write the function: Now that we have found C, we can write the function f(x) as: f(x) = 4/x + 2Evaluate at x = 1: Finally, we evaluate f(x) at x = 1 to find f(1). f(1) = 4/1 + 2 f(1) = 4 + 2 f(1) = 6

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$f^{\prime}(x)=-\frac{4}{x^{2}}$ and $f(2)=4$ $\newline$ $f(1)=$

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Algebra 1

Multiplication with rational exponents

Full solution

f^{\prime}(x)=-\frac{4}{x^{2}}

and

f(2)=4

\newline

f(1)=

Given derivative and point: We are given the derivative of a function $f(x)$ , which is $f'(x) = -\frac{4}{x^2}$ , and we are also given a point on the function, $f(2) = 4$ . To find $f(1)$ , we need to integrate the derivative $f'(x)$ to get $f(x)$ and then evaluate $f(x)$ at $x = 1$ .
Integrate the derivative: First, let's integrate the derivative $f'(x) = -\frac{4}{x^2}$ . The integral of $-\frac{4}{x^2}$ with respect to $x$ is $\frac{4}{x} + C$ , where $C$ is the constant of integration. $\newline$ $\int f'(x) \, dx = \int -\frac{4}{x^2} \, dx = \frac{4}{x} + C$
Find constant of integration: Now we need to find the constant of integration $C$ . We can do this by using the given point $f(2) = 4$ . Let's plug $x = 2$ into the integrated function to solve for $C$ . $\frac{4}{2} + C = 4$ $2 + C = 4$ $C = 4 - 2$ $C = 2$
Write the function: Now that we have found $C$ , we can write the function $f(x)$ as: $\newline$ $f(x) = \frac{4}{x} + 2$
Evaluate at $x = 1$ : Finally, we evaluate $f(x)$ at $x = 1$ to find $f(1)$ .
$f(1) = \frac{4}{1} + 2$
$f(1) = 4 + 2$
$f(1) = 6$