Question

verify that the given point is on the curve. find the lines that are (a) tangent and (b) normal to the curve at the given point.
6xy + π sin y = 61π, (4, 5π/2)
the point is on the curve because when is substituted for x and is substituted for y, the resulting statement is = 61π, which is a false statement.
(type an exact answer, using π as needed.)

Explanation:

Step1: Substitute x and y values

Substitute \(x = 4\) and \(y=\frac{5\pi}{2}\) into the left - hand side of the equation \(6xy+\pi\sin y\).
\[6\times4\times\frac{5\pi}{2}+\pi\sin\frac{5\pi}{2}\]

Step2: Simplify the first term

\[6\times4\times\frac{5\pi}{2}=60\pi\]

Step3: Simplify the second term

Since \(\sin\frac{5\pi}{2}=\sin(2\pi + \frac{\pi}{2})=\sin\frac{\pi}{2}=1\), then \(\pi\sin\frac{5\pi}{2}=\pi\).

Step4: Find the sum of the two terms

\[60\pi+\pi = 61\pi\]

Answer:

The point is on the curve because when \(4\) is substituted for \(x\) and \(\frac{5\pi}{2}\) is substituted for \(y\), the resulting statement is \(61\pi=61\pi\), which is a true statement.