given ( y = sin 2x ), ( \frac{dy}{dx} ) at ( x = 3 ) is most nearly\n*calculator required*\nr. 0.9600\nl…
given ( y = sin 2x ), ( \frac{dy}{dx} ) at ( x = 3 ) is most nearly\n*calculator required*\nr. 0.9600\nl. 0.9945\ng. 1.920\ni. 1.989
Answer
Explanation:
Step1: Differentiate the function
Using the chain rule, if (y = \sin(u)) and (u = 2x), then (\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}). Since (\frac{d}{du}(\sin(u))=\cos(u)) and (\frac{d}{dx}(2x) = 2), we have (\frac{dy}{dx}=2\cos(2x)).
Step2: Substitute (x = 3) into the derivative
Substitute (x = 3) into (\frac{dy}{dx}=2\cos(2x)). So (\frac{dy}{dx}\big|_{x = 3}=2\cos(6)). Using a calculator (in radian mode), (\cos(6)\approx0.9600). Then (2\cos(6)\approx2\times0.9600 = 1.920).
Answer:
G. 1.920