Math  /  Calculus

QuestionSuppose that the population of wolves is initially 800 , and that the rate of change of the population is given by r(t)r(t), where tt is in years. What is the meaning of 610r(t)dt\int_{6}^{10} r(t) d t The rate of change of the number of wolves in year 10 The change in the number of wolves from year 6 to year 10 Thê rate of change of the number of wolves in year 6 The difference in the growth rate r(t)r(t) between year 6 and year 10 . The total number of wolves after 10 years.

Studdy Solution

STEP 1

What is this asking? If we know how fast the wolf population is changing at any given time, what does the integral of that rate of change tell us over a specific time period? Watch out! Don't confuse the *rate of change* with the *actual change* in the wolf population.
Also, remember that the integral gives us the *total change* over a period, not just the rate at a single point in time.

STEP 2

1. Understand the Rate of Change
2. Relate Rate of Change to Population Change
3. Interpret the Definite Integral

STEP 3

We're given r(t)r(t) as the *rate of change* of the wolf population.
This means r(t)r(t) tells us how fast the wolf population is increasing or decreasing at any given time tt (measured in years).

STEP 4

Imagine a tiny time interval, say from tt to t+Δtt + \Delta t.
The change in the wolf population during this small interval is approximately r(t)Δtr(t) \cdot \Delta t.
Why? Because the rate of change multiplied by the time interval gives us the approximate change in population.
Think of it like speed and distance: speed \cdot time = distance.

STEP 5

To get the *total change* in the wolf population over a larger interval, say from year 6 to year 10, we need to add up all these tiny changes.
This is exactly what the integral does!

STEP 6

The definite integral 610r(t)dt\int_{6}^{10} r(t) dt represents the *total accumulated change* in the wolf population between year 6 and year 10.
It's the sum of all the tiny changes in population over that time period.

STEP 7

It's *not* the rate of change at a specific year (like year 6 or year 10).
It's the *total change* over the entire interval from year 6 to year 10.
It's also *not* the total number of wolves after 10 years, since we don't know how many wolves there were at year 6.
We only know the change in population between year 6 and 10.

STEP 8

The integral 610r(t)dt\int_{6}^{10} r(t) dt represents the change in the number of wolves from year 6 to year 10.

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