Question

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)

Accepted Answer

Recall the formula for the average rate of change of a function $$ f $$ from $$ x_1  to$$ $$ x_2 $$:

$$	ext{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}$$ Identify the function $$ f(x) = \sin x $$, and the given points $$ x_1 = \frac{5\pi}{4} $$ and $$ x_2 = \frac{3\pi}{2} . $$Calculate $$ f(x_1) = \sin \left( \frac{5\pi}{4} ight) . $$Recall that $$ \sin \left( \frac{5\pi}{4} ight) = -\frac{\sqrt{2}}{2} $$ because $$ \frac{5\pi}{4}  is in $$the third quadrant where sine is negative. Calculate $$ f(x_2) = \sin \left( \frac{3\pi}{2} ight) . $$Recall that $$ \sin \left( \frac{3\pi}{2} ight) = -1 $$ because it corresponds to the point at the bottom of the unit circle. Substitute these values into the average rate of change formula:

$$\frac{f\left( \frac{3\pi}{2} ight) - f\left( \frac{5\pi}{4} ight)}{\frac{3\pi}{2} - \frac{5\pi}{4}} = \frac{-1 - \left(-\frac{\sqrt{2}}{2}ight)}{\frac{3\pi}{2} - \frac{5\pi}{4}}$$

Simplify the numerator and denominator separately to find the exact average rate of change.

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)

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Key Concepts

Average Rate of Change

Sine Function Values at Special Angles

Simplifying Expressions Involving π