A large herd of reindeer is dying out. The number of...

A large herd of reindeer is dying out. The number of reindeer in the herd at time $t$ (measured in months), $0 \leq t \leq 15,$ is $$r(t)=25000-800 t-40 t^{2}-t^{3}$$ At what rate are the reindeer dying out after 11 months?

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Key Concepts

Differentiation of Polynomial Functions

Differentiation is a core technique in calculus used to find the derivative of a function. When applied to polynomial functions, it involves using rules such as the power rule to systematically compute how each term contributes to the overall rate of change.

Interpretation of Negative Rates

A negative derivative signifies that the quantity being measured is decreasing at that particular moment. This is an important aspect in many applications as it indicates a decline or loss over time, providing crucial information about the dynamic behavior of the system under study.

Instantaneous Rate of Change

This concept refers to the rate at which a quantity is changing at a specific moment. By taking the derivative of a function, one can determine how the function values change with respect to a small change in its input, thus providing insight into the speed and direction of change at any given point.

Transcript

00:01 We have a polynomial and it is supposed to model a population of reindeer.

00:08 So f of x equals a population.

00:14 And actually this should be f of t since we're using the variable t in the polynomial.

00:19 So let me just make that a little technical adjustment.

00:26 So f of t is modeling the population of reindeer.

00:36 Is time, t for time.

00:41 So now we don't necessarily have to graph this or plug in a lot of values to get a sense of what happens as we go through time.

00:56 So as t goes to infinity because time keeps increasing for infinity, never goes backwards, so we can use the leading coefficient test to tell us.

01:12 So first, what is the leading coefficient? well, first, we have to take a look at the power.

01:16 What's the biggest power? it's five.

01:19 So the biggest power is five.

01:20 So that puts us in the area of the leading coefficient test that is odd.

01:28 So we're here because five is an odd number.

01:31 And the leading coefficient then is negative 0 .125.

01:38 And which is negative.

01:41 So this meets in our table over here, where the graph rises to the left and falls to the right.

01:48 So then this is t time, and then this would be population...