The metabolic rate of a person who has just eaten a meal...

The metabolic rate of a person who has just eaten a meal tends to go up and then, after some time has passed, returns to a resting metabolic rate. This phenomenon is known as the thermic effect of food. Researchers have indicated that the thermic effect of food for one particular person is $$ F(t)=-10.28+175.9 t e^{-t / 1.3}, $$ where $F(t)$ is the thermic effect of food (in $\mathrm{kJ} / \mathrm{hr}$ ) and $t$ is the number of hours that have elapsed since eating a meal. a. Find $F^{\prime}(t)$. b. Determine where this function is increasing and where it is decreasing. Interpret your answers.

The metabolic rate of a person who has just eaten a meal tends to go up and then, after some time has passed, returns to a resting metabolic rate. This phenomenon is known as the thermic effect of food. Researchers have indicated that the thermic effect of food for one particular person is
$$
F(t)=-10.28+175.9 t e^{-t / 1.3},
$$
where $F(t)$ is the thermic effect of food (in $\mathrm{kJ} / \mathrm{hr}$ ) and $t$ is the number of hours that have elapsed since eating a meal.
a. Find $F^{\prime}(t)$.
b. Determine where this function is increasing and where it is decreasing. Interpret your answers.

Key Concepts

Increasing and Decreasing Functions

Determining where a function is increasing or decreasing involves analyzing the sign of its derivative. If the derivative is positive over an interval, the function is increasing, indicating that the metabolic rate is rising; if negative, the function is decreasing, showing a decline in the metabolic rate. This analysis is crucial in understanding the behavior of a system or model over time, such as the thermic effect of food in this scenario.

Chain Rule

The chain rule is a method for taking the derivative of composite functions, where one function is nested within another. In this problem, differentiating the exponential part involves the chain rule because the exponent itself is a function of time. This rule is vital for navigating through layers of functions to correctly calculate the overall derivative.

Product Rule

The product rule is a fundamental technique used in differentiation when dealing with functions that are products of two or more simpler functions. In the given problem, the metabolic rate function includes a term that is the product of a linear function and an exponential function, so the product rule is essential for correctly computing the derivative.

Derivative

A derivative represents the instant rate of change of a function with respect to a variable. In this context, finding the derivative of the metabolic rate function gives insight into how the body’s rate of energy expenditure changes over time after eating. It is a central concept in calculus, providing the slope of the tangent line to a function's graph at any given point.

Transcript

00:01 For this problem, we have been given an equation showing the thermic effect of food on a person.

00:07 And this function is in terms of t, where t is the number of hours that have elapsed since you've eaten a meal.

00:14 So we want to know where this function is increasing and where it is decreasing.

00:19 To do that, we need to start by finding the derivative.

00:22 So the derivative of this function is going to be 175 .9 times the derivative.

00:31 Of this piece that's in red.

00:33 So this is a product.

00:36 So it's going to be the first times the derivative of the second.

00:43 And i do have to multiply that by the derivative of that exponent.

00:47 So that's going to be times negative 1 over 1 .3 plus the second times the derivative of the first.

01:00 Okay.

01:01 Now they do have something in common.

01:03 So i can pull out in addition to the 175.

01:08 I can pull out e to the negative t divided by 1 .3.

01:13 And what i have left is negative t over 1 .3 plus 1.

01:21 So i'm going to box this in green.

01:23 I don't want to lose this.

01:24 This is the equation of our derivative.

01:27 Now, to know where our function is increasing and decreasing, we need to find where this derivative equation equals zero.

01:35 So that will tell us places where i could be switching from from increasing to decreasing or from decreasing to increasing.

01:43 So let's set this equal to zero.

01:45 Well, this piece here, e to anything, e to any exponent, will never equal zero.

01:52 So if i want this function to equal zero, the only piece that could possibly equal zero is this factor right here.

01:59 Negative t divided by 1 .3 plus 1...

Please give Ace some feedback