Can someone explain how find the average rate change for these functions?
1.) f(x)=x^3+1
a (2,3) b.(-1,1)
2.) h(t)= cot t
a. pie/4, 3pie/4 b. pie/6, pie/2
I am so lost. T_T please help!
Sure! I'll explain how to find the average rate change for these functions step by step.
1.) For the function f(x) = x^3 + 1, we want to find the average rate change between two given points.
a.) Given points (2,3): To find the average rate change between these points, we need to calculate the change in y-values (Δy) and the change in x-values (Δx). In this case, our Δy is 3 - 1 = 2, and Δx is 2 - (-1) = 3.
To find the average rate change, we divide Δy by Δx: Average rate change = Δy / Δx = 2 / 3.
Therefore, the average rate change between points (2,3) is 2/3.
b.) Given points (-1,1): Similarly, we need to calculate the change in y-values (Δy) and the change in x-values (Δx). This time, our Δy is 1 - 1 = 0, and Δx is -1 - (-1) = 0.
We find that the average rate change between points (-1,1) is 0/0. However, dividing by zero is undefined, so we can't calculate the average rate change at this point.
2.) For the function h(t) = cot(t), we also want to find the average rate change between two given points.
a.) Given points (π/4, 3π/4): Follow the same steps as before. First, find the change in y-values (Δy), which is cot(3π/4) - cot(π/4). Then, find the change in x-values (Δx), which is (3π/4) - (π/4).
You'll get Δy = -1 - 1 = -2, and Δx = π/2. Finally, find the average rate change: Average rate change = Δy / Δx = -2 / (π/2).
Therefore, the average rate change between points (π/4, 3π/4) is -4/π.
b.) Given points (π/6, π/2): Again, calculate the change in y-values (Δy), which is cot(π/2) - cot(π/6), and the change in x-values (Δx), which is π/2 - π/6.
You'll find Δy = 0 - √3/3, and Δx = π/3. Calculate the average rate change: Average rate change = Δy / Δx = -(√3/3) / (π/3).
Therefore, the average rate change between points (π/6, π/2) is -√3/π.
These calculations will give you the average rate change for the given functions and points. Remember that the average rate change measures how much a function is changing on average over a specific interval.