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Problem 536
536
The equation of a curve is \(y = 3 \cos 2x\). The equation of a line is \(x + 2y = \pi\). On the same diagram, sketch the curve and the line for \(0 \leq x \leq \pi\).
Solution
The curve \(y = 3 \cos 2x\) is a cosine function with amplitude 3 and period \(\pi\). It completes one full oscillation from \(x = 0\) to \(x = \pi\). The range of the curve is from -3 to 3.
The line \(x + 2y = \pi\) can be rewritten as \(y = \frac{\pi - x}{2}\). This is a straight line with a negative slope of -1/2 and a y-intercept of \(\frac{\pi}{2}\).
To sketch the curve, plot the cosine wave starting at \(y = 3\) when \(x = 0\), decreasing to \(y = -3\) at \(x = \frac{\pi}{2}\), and returning to \(y = 3\) at \(x = \pi\).
To sketch the line, plot the intercepts: when \(x = 0\), \(y = \frac{\pi}{2}\); when \(y = 0\), \(x = \pi\). Draw the line through these points.
