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Chapter 2: Problem 88

Rate of Change Determine whether there exist any values of \(x\) in the interval \([0,2 \pi)\) such that the rate of change of \(f(x)=\sec x\) and the rate of change of \(g(x)=\csc x\) are equal.

Short Answer

Expert verified
The values of \(x\) in the interval \( [0,2 \pi) \) such that the rate of change of \(f(x)=sec x\) and the rate of change of \(g(x)=csc x\) are equal are \(x = \frac{\pi}{4}\) and \(x = \frac{3\pi}{4}\).

Step by step solution

01

Find the derivative of f(x)

The derivative of \(f(x) = sec(x)\) is \(f'(x) = sec(x)tan(x)\). This is the rate of change of \(f(x)\).
02

Take the derivative of g(x)

The derivative of \(g(x) = csc(x)\) is \(g'(x) = -csc(x)cot(x)\). This is the rate of change of \(g(x)\).
03

Set the derivatives equal

Next, we set \(f'(x) = g'(x)\), thereby getting the equation \(sec(x)tan(x) = -csc(x)cot(x)\). This allows us to solve for the common values of \(x\) in the given interval.
04

Solve the equation

Using properties of trigonometric functions, this equation simplifies to \(-tan^2(x) -1 = 0\). Solving this equation, we find \(x = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}\). However, only \(\frac{\pi}{4}\) and \(\frac{3\pi}{4}\) lie in the interval \([0,2\pi)\), so these are the solutions.

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