can someone also help me ndertsand this question ; y=x^3 +2x^2+3 represents the rate of an object over the interval [7, 10]. What is the average rate of the object over the time interval?
find y at the end (x = 10)
find y at the beginning (x=3)
subtract
Yend - Ybegin
divide that by 10-7 which is 3
actually, the above solution applies if y is the position function. Since it is itself a rate, you need the average value of the function. That is
∫[7,10] x^3+2x^2+3 dx
----------------------------- = 782.25
10-7
To find the average rate of the object over the time interval [7, 10], we can use the concept of average rate of change.
The average rate of change represents how much a quantity (in this case, the object's rate) changes on average per unit of the input (in this case, time).
In this question, we are given the function y = x^3 + 2x^2 + 3, which represents the rate of the object. To find the average rate of the object over the interval [7, 10], we need to calculate the change in the object's rate and divide it by the change in time over that interval.
To do this, follow these steps:
Step 1: Substitute the values of the interval endpoints into the function.
- For the lower endpoint (7), substitute it into the function: y(7) = 7^3 + 2(7)^2 + 3.
- For the upper endpoint (10), substitute it into the function: y(10) = 10^3 + 2(10)^2 + 3.
Step 2: Calculate the difference in the object's rate between the two endpoints.
- Subtract the value of y(7) from y(10): y(10) - y(7).
Step 3: Calculate the difference in time between the two endpoints.
- Subtract the lower endpoint (7) from the upper endpoint (10): 10 - 7.
Step 4: Divide the change in the object's rate by the change in time to get the average rate of the object.
- Divide the result from Step 2 by the result from Step 3: (y(10) - y(7)) / (10 - 7).
By following these steps, you should be able to find the average rate of the object over the given time interval.