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Differentiation is a fundamental concept in calculus focused on determining how a function changes at any given point. When you differentiate a function, you're essentially finding a new function that gives the slope or rate of change of the original function. In the context of our problem, the concentration of medication over time is given by the function \( C(t) = 10 t^2 e^{-t} \). By differentiating this function, we are finding how fast or slow the concentration changes over time.

To differentiate \( C(t) \), we apply the product and chain rules of differentiation. Specifically, we have the product of two functions: \( 10t^2 \) and \( e^{-t} \). The derivative is computed as:

• \( \frac{d}{dt}[10t^2]e^{-t} + 10t^2 \frac{d}{dt}[e^{-t}] \) which gives us \( 2t \, 10e^{-t} \) and \( -t^2 \, 10e^{-t} \), respectively.
• Combining results in: \( C'(t) = 10e^{-t}(2t - t^2) \).

The derivative \( C'(t) \) is then used to analyze how the concentration behaves at various points in time, showing both increasing and decreasing trends.

Graph sketching involves plotting the main features of a function to understand its behavior visually. For functions like \( C(t) = 10 t^2 e^{-t} \), the graph displays how the concentration of the medication changes over time.

To sketch the graph, you plot specific points that were already calculated:

• \((0,0)\): The concentration starts at zero.
• \((1,3.68)\): Concentration increases.
• \((2,5.41)\): The concentration peaks.
• \((3,4.48)\): Begins to decrease.
• \((10,0.045)\): Close to zero reflects the exponential decay.

By connecting these points smoothly, the graph starts from the origin, rises to the maximum at \(t = 2\), and then gradually decreases, illustrating the medication being absorbed and then reducing as it's metabolized. Graph sketching is a helpful tool to visually interpret how functions like these behave over time.

Exponential functions are characterized by a constant raised to the power of a variable. They have unique properties, including growth or decay, depending on whether the exponent is positive or negative. In the function \( C(t) = 10 t^2 e^{-t} \), the \( e^{-t} \) component represents exponential decay, where the base \( e \) is Euler's number (approximately 2.718).

Exponential decay implies that as \( t \) increases, \( e^{-t} \) decreases rapidly towards zero. This property is why, in the context of our exercise, the concentration starts decreasing after a certain point (after peaking). The initial multiplier \( 10t^2 \) influences how quickly the concentration increases before the effect of decay becomes dominant.

Understanding exponential functions helps in problems involving continuous growth or decay, like population models, radioactive decay, and in this case, concentration levels in a biological system.