Let r'(t) represent the rate of growth of a dog in pounds per year. What does r(t) represent?
What does integral of 2 to 6 r'(t) represent about the dog?
weight of dog at time t
how much weight the dog gained between those times
Well, r(t) is the rate at which the dog's weight is changing over time. So, if r'(t) represents the rate of growth in pounds per year, then r(t) simply represents the rate of change in the dog's weight at any given time.
As for the integral of 2 to 6 r'(t), that would be the total change in weight of the dog between years 2 and 6. It's like adding up all those little weight changes over that time period. So, if the integral is positive, it means the dog gained weight. If it's negative, well, the poor pup was on a diet and lost weight. So, it tells us about the overall weight change of the dog during those years. I hope it didn't go off the leash and get into too many treats!
r(t) represents the instantaneous rate of growth of the dog in pounds per year at a specific time t.
The integral of 2 to 6 r'(t) represents the total change in the weight of the dog between the time intervals 2 and 6. In other words, it represents the overall increase or decrease in weight of the dog from time t=2 to t=6.
To understand what r(t) represents, let's break down the terms:
- "r'(t)" represents the derivative of some function "r(t)" with respect to time "t." In this context, it specifically represents the rate of growth of a dog in pounds per year. The prime notation (') is commonly used to denote the derivative of a function.
Now, let's move on to interpreting the integral:
- The integral represents the accumulation of a quantity over a given interval. In this case, the integral of "r'(t)" from 2 to 6 represents the accumulation of the rate of growth of the dog over the time interval from time t = 2 to t = 6.
To further understand what this integral signifies about the dog, we can relate it to real-world context. If we assume that "r(t)" represents the weight of the dog at time t, then "r'(t)" represents the rate at which the weight is changing at that particular time.
By integrating "r'(t)" over the time interval from 2 to 6, we find the total change in the weight of the dog during that time period. This integral value will give us the accumulated growth of the dog, expressed in pounds, from 2 to 6 years.