The derivative of the function f is defined by f^(')(x)=(x^(2)+1)cos(2x+1). If f(1)=1, then use a calculator to find the value of f(6) to the nearest thousandth. Answer:

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The derivative of the function  f is defined by  f^(')(x)=(x^(2)+1)cos(2x+1). If  f(1)=1, then use a calculator to find the value of  f(6) to the nearest thousandth. Answer:

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Integrate f'(x): To find f(6), we need to integrate the derivative f'(x) to get the original function f(x). The integral of f'(x) will give us f(x) + C, where C is the constant of integration.Numerical Approximation: We integrate f'(x) = (x^2 + 1)cos(2x + 1). This requires integration by parts or a substitution method. However, this integral does not have an elementary antiderivative, so we cannot express f(x) in terms of elementary functions. Instead, we will need to use numerical methods to approximate the integral from 1 to 6.Initial Value: We know that f(1) = 1. This means that when we integrate f'(x) from 1 to 6, we need to add the initial value f(1) to our result to find f(6).Perform Numerical Integration: We use numerical integration (like the trapezoidal rule, Simpson's rule, or a numerical integration function on a calculator) to approximate the integral of f'(x) from 1 to 6. This step requires a calculator or computer software and cannot be done analytically.Add Result and Initial Value: After performing the numerical integration, we add the result to the initial value f(1) = 1 to find the value of f(6).Round Approximate Value: Assuming the numerical integration was done correctly and we added the initial value f(1) = 1, we now have an approximate value for f(6). We round this value to the nearest thousandth as requested.

The derivative of the function $f$ is defined by $f^{\prime}(x)=\left(x^{2}+1\right) \cos (2 x+1)$ . If $f(1)=1$ , then use a calculator to find the value of $f(6)$ to the nearest thousandth. $\newline$ Answer:

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The derivative of the function f f f is defined by f′(x)=(x2+1)cos⁡(2x+1) f^{\prime}(x)=\left(x^{2}+1\right) \cos (2 x+1) f′(x)=(x2+1)cos(2x+1). If f(1)=1 f(1)=1 f(1)=1, then use a calculator to find the value of f(6) f(6) f(6) to the nearest thousandth.\newlineAnswer:

The derivative of the function $f$ is defined by $f^{\prime}(x)=\left(x^{2}+1\right) \cos (2 x+1)$ . If $f(1)=1$ , then use a calculator to find the value of $f(6)$ to the nearest thousandth. $\newline$ Answer: