8. let f be a differentiable function such that f(0) = 2 and f(x) = sin x. what is the value of f(π)? again…
8. let f be a differentiable function such that f(0) = 2 and f(x) = sin x. what is the value of f(π)? again, please compute the answer by hand.
Answer
Explanation:
Step1: Recall the fundamental theorem of calculus
By the fundamental theorem of calculus, $f(x)-f(0)=\int_{0}^{x}f^{\prime}(t)dt$.
Step2: Substitute the given values
We know that $f(0) = 2$ and $f^{\prime}(x)=\sin x$, so $f(x)=2+\int_{0}^{x}\sin tdt$.
Step3: Evaluate the integral
We know that $\int\sin tdt=-\cos t + C$. Then $\int_{0}^{x}\sin tdt=-\cos x-(-\cos0)=-\cos x + 1$.
Step4: Find $f(\pi)$
Substitute $x = \pi$ into $f(x)$. So $f(\pi)=2+(-\cos\pi + 1)$. Since $\cos\pi=-1$, then $f(\pi)=2+( -(-1)+1)=4$.
Answer:
$4$