Let f(x) = sin x, g(x) = cos x, and h(x) = 2x. Find the exact value of each expression. Do not use a calculator. the average rate of change of f from x₁ = 5𝜋/4 to x₂ = 3𝜋/2 (Hint: the average rate of change of f from x₁ to x₂ is f(x₂) - f(x₁)/(x₂ - x₁)

Verified step by step guidance

1. Recall the formula for the average rate of change of a function \( f \) from \( x_1 \) to \( x_2 \): \[\text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}\]
2. Identify the function \( f(x) = \sin x \), and the given points \( x_1 = \frac{5\pi}{4} \) and \( x_2 = \frac{3\pi}{2} \).
3. Calculate \( f(x_1) = \sin \left( \frac{5\pi}{4} \right) \). Recall that \( \sin \left( \frac{5\pi}{4} \right) = -\frac{\sqrt{2}}{2} \) because \( \frac{5\pi}{4} \) is in the third quadrant where sine is negative.
4. Calculate \( f(x_2) = \sin \left( \frac{3\pi}{2} \right) \). Recall that \( \sin \left( \frac{3\pi}{2} \right) = -1 \) because it corresponds to the point at the bottom of the unit circle.
5. Substitute these values into the average rate of change formula: \[\frac{f\left( \frac{3\pi}{2} \right) - f\left( \frac{5\pi}{4} \right)}{\frac{3\pi}{2} - \frac{5\pi}{4}} = \frac{-1 - \left(-\frac{\sqrt{2}}{2}\right)}{\frac{3\pi}{2} - \frac{5\pi}{4}}\] Simplify the numerator and denominator separately to find the exact average rate of change.

Key Concepts

Average Rate of Change

The average rate of change of a function between two points measures how much the function's output changes per unit change in input. It is calculated as the difference in function values divided by the difference in input values, representing the slope of the secant line connecting the points.

Sine Function Values at Special Angles

The sine function has well-known exact values at special angles such as π/4 and π/2. For example, sin(5π/4) = -√2/2 and sin(3π/2) = -1. Knowing these values allows for exact computation without a calculator.

Simplifying Expressions Involving π

When working with trigonometric functions and intervals involving π, it is important to handle the subtraction and simplification of expressions like (3π/2 - 5π/4) carefully. This ensures accurate calculation of intervals and rates of change.