Joe is investigating the rate of change of the function y=cos x on the interval xE[0,2π]. He notices that the graph of y=cos x passes through the x-axis at 45°. He also determines the instantaneous rate of change at x = 0, π, and 2π by inspection. Based on this information, determine an equation r(x) to predict the instantaneous rate of change of the function y=cos x on the interval xE[0,2π]. Then use the equation to calculate the exact instantaneous rate of change at x = π/4.

Thanks!

Why did the cosine go to the doctor?

Because it had a sinus infection!

To determine an equation, r(x), that predicts the instantaneous rate of change of the function y = cos(x) on the interval x ∈ [0, 2π], we first need to understand the key properties of the function.

1. The graph of y = cos(x) passes through the x-axis at 45°. This means that cos(45°) = 0, which corresponds to x = π/4 in radians.

2. We need to determine the instantaneous rate of change at x = 0, π, and 2π by inspection.

- At x = 0, we can see that the value of cos(0) is 1. This means that the instantaneous rate of change at x = 0 is positive.

- At x = π, we observe cos(π) = -1. This means that the instantaneous rate of change at x = π is negative.

- At x = 2π, we again see that cos(2π) = 1. Therefore, the instantaneous rate of change at x = 2π is positive.

Based on the information above, we can infer that the instantaneous rate of change of y = cos(x) follows a pattern of positive, negative, positive, negative, and so on. To predict the pattern, we can use the concept of period and the fact that the cosine function repeats every 2π radians.

The equation for r(x) can be written as:

r(x) = (-1)^(n+1)

Where n = floor(x/π). This equation predicts the instantaneous rate of change based on the pattern we observed.

To calculate the exact instantaneous rate of change at x = π/4, we substitute π/4 into the equation:

r(π/4) = (-1)^(floor(π/4π + 1))

Since π/4π + 1 simplifies to 1.25, we have:

r(π/4) = (-1)^(1.25)

Using the fact that any real number raised to a power of 1.25 can be written as the square root of the number raised to the power of 5/4, we can simplify further:

r(π/4) = (sqrt(-1))^5

The square root of -1 is represented by the imaginary unit, denoted as i. Therefore:

r(π/4) = i^5

Using the property of imaginary powers, we know that i^4 = 1. Therefore, we can rewrite the expression:

r(π/4) = i^4 * i

Since i^4 = 1, we have:

r(π/4) = 1 * i

Finally, the exact instantaneous rate of change at x = π/4 is given by r(π/4) = i.

Please note that in this context, the term "rate of change" refers to the derivative of the function y = cos(x), which represents the slope of the tangent line to the graph of y = cos(x) at a specific x value.

To determine the equation r(x) that predicts the instantaneous rate of change of the function y=cos x on the interval [0,2π], we need to find the derivative of the function. The derivative represents the rate at which the function is changing at any given point.

The derivative of y=cos x can be found using the chain rule. The derivative of cos x is -sin x, so we have:

r(x) = -sin x

Now, we can use this equation to calculate the exact instantaneous rate of change at x = π/4.

Substituting x = π/4 into the equation r(x) = -sin x, we get:

r(π/4) = -sin(π/4)

Calculating the value, we have:

r(π/4) = -√2/2

Therefore, the exact instantaneous rate of change at x = π/4 is -√2/2.