Thermic Effect of Food As we saw in the last section, the...

Thermic Effect of Food As we saw in the last section, the metabolic rate after a person eats 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 and can be described for a particular individual as $$F(t)=-10.28+175.9 t e^{-t / 1.3}$$ 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. Find the time after the meal when the thermic effect of the food is maximized. Source: American Journal of Clinical Nutrition.

Thermic Effect of Food As we saw in the last section, the metabolic rate after a person eats 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 and can be described for a particular individual as
$$F(t)=-10.28+175.9 t e^{-t / 1.3}$$
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. Find the time after the meal when the thermic effect of the food is maximized. Source: American Journal of Clinical Nutrition.

Key Concepts

Function Maximization

This concept involves finding the highest or lowest point (specifically the maximum or minimum value) of a function. In calculus, it typically requires taking the derivative of the function, identifying where that derivative is zero or undefined (critical points), and then determining which of these points yields the largest or smallest function value. This is essential in many applications, including optimizing biological metrics, economics, and physics problems.

Critical Point Analysis and Derivatives

Critical point analysis uses derivatives to identify points on a function where the slope is zero or the derivative does not exist. For smooth functions, these points are potential candidates for local maxima or minima. The process involves calculating the first derivative, setting it equal to zero, and solving for the variable. This method is fundamental in determining optimal values in various functions, including those with applied physical significance.

Product Rule and Chain Rule

When differentiating functions that are products of two or more component functions or include composite functions such as exponential expressions, the product rule and chain rule are required. The product rule is used when differentiating the product of two functions, while the chain rule is applied to differentiate composite functions. Mastery of these rules is crucial for successfully finding derivatives needed for optimization tasks.

Exponential Function Behavior

Exponential functions, particularly those representing decay or growth, have distinct properties where the rate of change is proportional to the function's current value. In optimization problems, understanding this behavior is important when the function includes an exponential term, as it affects the location and nature of the maximum or minimum. This concept is widely used in modeling scenarios in fields like biology, physics, and economics.

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