given the graph of y=cos θ + pi/2 from 0≤ θ ≤ 2pi,
a) for what values of θ does the instantaneous rate of change appear to equal 0? (I said pi/2 and 3pi/2)
b) for what values of θ does the instantaneous rate of change reach its maximum? its minimum? (this part I don't get)
If you wrote what you meant, then y' = -sinθ y'=0 at θ=0,π,2π
If you really meant y = cos(θ + π/2) = -sinθ, then y' = -cosθ, so y'=0 at θ = π/2, 3π/2 as you said.
so, using the 2nd version, -cosθ achieves its maximum at θ=π, minimum at 0,2π
You can always check the graph for confirmation.
The instantaneous rate of change is your first derivative which would be
dy/dθ = -sinθ
if -sinθ = 0, θ = 0, π, and 2π for your given interval
if your initial equation is y = cos (θ + π/2) , then
dy/dθ = -sin(θ+π/2)
and for -sin(θ+π/2) = 0, θ = π/2, 3π/2
That was your answer, so the brackets ARE NECESSARY
In that case for the first derivative to have a max, its derivative, namely the
2nd derivative of the original must be zero.
then y '' = -cos(θ+π/2) = 0 would be for the function with brackets.
θ = 0, π, 2π
you decide which way the original equation was
sorry I forgot the brackets. I did mean cos(θ + π/2)
To find the values of θ for which the instantaneous rate of change of y = cos(θ) + π/2 appears to equal zero, we can start by analyzing the graph and identifying the points where the slope seems to be zero.
a) Analysis of the given function:
The graph of y = cos(θ) + π/2 is a sinusoidal wave where the cosine function is shifted vertically upward by π/2 units. It has a maximum value of 1 and a minimum value of 0.
To find the values of θ where the instantaneous rate of change appears to be zero, we need to identify the points where the graph has horizontal tangents. These points are usually found at the maximum and minimum values of the function.
From the given equation y = cos(θ) + π/2, we know that the maximum value of y is π/2 + 1, and the minimum value is π/2 + 0.
So, the maximum values occur when cos(θ) = 1, and the minimum values occur when cos(θ) = 0.
Cosine function reaches its maximum (cos(θ) = 1) at θ = 0 and θ = 2π (or 0° and 360° in degrees).
Cosine function reaches its minimum (cos(θ) = 0) at θ = π/2 and θ = 3π/2 (or 90° and 270° in degrees).
Therefore, for the given range 0 ≤ θ ≤ 2π, the values of θ where the instantaneous rate of change appears to equal zero are π/2 and 3π/2.
b) To find the values of θ where the instantaneous rate of change reaches its maximum or minimum, we need to look at the behavior of the graph around those points.
Since the instantaneous rate of change represents the slope of the graph at a given point, the values of θ where the slope is steepest would correspond to maximum and minimum rates of change.
The slope is steepest (maximum rate of change) when the graph is transitioning from an increasing trend to a decreasing trend or vice versa. This occurs at the maximum and minimum values of the function.
In the given equation, the maximum value of y is π/2 + 1, and the minimum value is π/2 + 0.
Therefore, the values of θ where the instantaneous rate of change reaches its maximum are θ = 0 and θ = 2π.
Similarly, the values of θ where the instantaneous rate of change reaches its minimum are θ = π/2 and θ = 3π/2.
Note: The explanation given is based on the pattern and characteristics of the cosine function and might not take into account specific fine details of the graph.