Let f(t) and g(t) give, respectively, the amount of water...

Let $f(t)$ and $g(t)$ give, respectively, the amount of water (in acre-feet) in two different reservoirs on day $t$ Suppose that $f(0)=2000, g(0)=1500$ and that $f^{\prime}(0)=11, g^{\prime}(0)=13.5 .$ Let $h(t)=f(t)-g(t)$ (a) Evaluate $h(0)$ and $h^{\prime}(0) .$ What do these quantities tell you about the reservoir? (b) Assume $h^{\prime}$ is constant for $0 \leq t \leq 250 .$ Does $h$ have any zeros? What does this tell you about the reservoirs?

Let $f(t)$ and $g(t)$ give, respectively, the amount of water (in acre-feet) in two different reservoirs on day $t$ Suppose that $f(0)=2000, g(0)=1500$ and that $f^{\prime}(0)=11, g^{\prime}(0)=13.5 .$ Let $h(t)=f(t)-g(t)$
(a) Evaluate $h(0)$ and $h^{\prime}(0) .$ What do these quantities tell you about the reservoir?
(b) Assume $h^{\prime}$ is constant for $0 \leq t \leq 250 .$ Does $h$ have any zeros? What does this tell you about the reservoirs?

Key Concepts

Difference Function

This concept involves defining a new function as the difference between two given functions. By subtracting one function from another, you create a function that quantifies the relative difference between the two quantities being measured. In various contexts, this is used to compare two related processes or quantities, such as the amounts of water in two reservoirs, highlighting how one lags behind or exceeds the other over time.

Instantaneous Rate of Change (Derivative)

The derivative of a function represents its instantaneous rate of change at a given point. It provides insight into how the function's value is changing at that moment. When analyzing two functions, taking the derivative of their difference, as in the given problem, allows one to assess the relative rates at which the underlying quantities are increasing or decreasing, offering an understanding of their dynamic behavior.

Linear Behavior from Constant Derivative

If a function’s derivative remains constant over an interval, the function itself is linear over that interval. This means the function changes at a steady rate. In the context of the problem, assuming a constant derivative for the difference function implies that the relative difference between the two quantities evolves linearly with time, making it straightforward to predict future values and analyze when the difference might reach certain thresholds.

Zeros of a Function (Roots)

Finding the zeros of a function involves solving for the points where the function equals zero. These zeros are significant because they often represent break-even points or moments of transition. In comparative contexts, such as when comparing two reservoirs, a zero of the difference function indicates the time at which the two quantities are equal, which can have important practical implications.

Transcript

00:01 So we're given some values of f of t and g of t, which just give us the amount of water in two different reservoirs.

00:06 And we're also given their derivatives at the same point zero.

00:11 And we're also given this function h of t, which is just the difference between f of t and g of t.

00:18 So for part a, we're trying to find h of zero.

00:22 And this is just equal to f of zero minus g of zero.

00:26 And if we look at our values, that's equal to 2 ,000 minus 1 ,500, which is equal to 500.

00:36 So the difference between the function that is represented by f of t and g of t, or sorry, the reservoir that's represented by f of t and the reservoir that's represented by g of t is 500 acre feet of water.

00:50 And so now we're going to find h prime of 0, and this is just equal to f prime.

00:57 Of 0 minus g prime of 0 and so again we were given these values and they were 11 and 13 .5 so this is equal to negative 2 .5 and now for part b we're told that h prime of t is equal to some constant for 0 to t to 250 so from t is equal to 0 to t is equal to to 250, we have some constant as our derivative.

01:32 And we know that h prime of zero is equal to negative 2 .5.

01:35 So that means that our constant a is equal to negative 2 .5.

01:40 And what this means is that our slope is going to be a constant negative 2 .5 slope from 0 to 250.

01:48 And i can show you, or i can give you an example to try and illustrate this...

Please give Ace some feedback