Researchers have developed the following function that can...

Researchers have developed the following function that can be used to accurately predict the weight of Holstein cows (females) of various ages: $$W_{1}(t)=619\left(1-0.905 e^{-0.002 t}\right)^{1.2386}$$ where $W_{1}(t)$ is the weight of the Holstein cow (in kilograms) that is $t$ days old. Where is this function increasing?

Researchers have developed the following function that can be used to accurately predict the
weight of Holstein cows (females) of various ages:
$$W_{1}(t)=619\left(1-0.905 e^{-0.002 t}\right)^{1.2386}$$
where $W_{1}(t)$ is the weight of the Holstein cow (in kilograms) that is $t$ days old. Where is this function increasing?

Key Concepts

Function Domain

The domain of a function is the set of all input values for which the function is defined. In age-based models like this, the domain is typically restricted to non-negative values (t ? 0 days). Understanding the domain is crucial for correctly interpreting where the function’s increasing behavior is applicable.

Monotonicity

Monotonicity is the property of a function to be either entirely non-decreasing or non-increasing over an interval. In the context of the problem, determining where the weight function is increasing involves checking that every increase in the age variable leads to an increase in weight, reflecting a consistently growing pattern.

Derivative Test

The derivative test is a fundamental concept in calculus used to determine the monotonic behavior of a function. By taking the derivative of the function and analyzing its sign, one can identify intervals where the function is increasing (derivative positive) or decreasing (derivative negative). This approach is essential in pinpointing regions of increase for the weight function.

Exponential Decay

Exponential decay describes a process where a quantity decreases at a rate proportional to its current value. In the weight prediction function, the term involving e^(–0.002t) represents an exponential decay component, which, when subtracted from one and then raised to a power, is transformed into a component that contributes to the overall increasing behavior of the function.

Function Transformation

Function transformation involves applying operations such as scaling, translating, and exponentiating to a basic function in order to model more complex behaviors. The given function transforms an exponential decay term through scaling, subtraction, and exponentiation to accurately represent the biological growth process of Holstein cows over time.

Transcript

00:01 So we have been given this function that shows the weight of female holstained cows at day t, you know, t being the number of days since they've been born.

00:11 And we'd like to know when is this function increasing and when is it decreasing.

00:18 So in order to find increasing and decreasing intervals, our first step is to find the derivative.

00:23 So if i want the derivative of this function, w sub 1 of t, i start 619 is a constant.

00:33 It would be 619 times.

00:35 Well, the derivative of this piece here, i'm going to pull down the 1 .236 times what's in parentheses.

00:49 And i subtract 1 from my exponent times the derivative of what's inside the parentheses.

00:57 So times what's going to be negative 0 .905, e to the negative 0 .002t times times.

01:08 The derivative of that exponent, which is negative 0 .002.

01:14 Okay.

01:16 Let's simplify this slightly.

01:17 We have a lot of constants that we can put together.

01:21 I am multiplying by 619, 1 .236, negative 0 .9, and negative 0 .002.

01:31 Multiplying those all together gives me 1 .3877 times, well, i have an e here, so it's times e to the negative 0 .002t, times my parentheses, 1 minus 0 .905, e to the negative 0 .002t, raised to the 0 .236 exponent.

02:01 Okay, so here's my equation.

02:04 This is the derivative of this function.

02:07 To find where i could be changing from increasing to decrease, or from decreasing back to increasing, i need to know when is the derivative function equal to zero.

02:18 So i want to set this equal to zero and see what values of t that gives me.

02:23 Well, let's examine this piece by piece.

02:27 My constant, and i'll just kind of use red as i walk through here, my constant is positive, so that's not going to give me a zero value.

02:35 E to any power is never going to give me zero.

02:39 It's always going to be positive.

02:40 So i'm not going to get a zero from here.

02:43 So the only way i could get a zero would be for this to equal zero...

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