In a medical experiment, the body weight of a baby rat in...

In a medical experiment, the body weight of a baby rat in the control group after $t$ days was $f(t)=4.96+.48 t+.17 t^{2}-.0048 t^{3}$ grams. (Source: Growth, Development and Aging.) (a) Graph $f(t)$ in the window $[0,20]$ by $[-12,50] .$ (b) Approximately how much did the rat weigh after 7 days? (c) Approximately when did the rat's weight reach 27 grams? (d) Approximately how fast was the rat gaining weight after 4 days? (e) Approximately when was the rat gaining weight at the rate of 2 grams per day? (f) Approximately when was the rat gaining weight at the fastest rate?

In a medical experiment, the body weight of a baby rat in the control group after $t$ days was $f(t)=4.96+.48 t+.17 t^{2}-.0048 t^{3}$ grams. (Source: Growth, Development and Aging.)
(a) Graph $f(t)$ in the window $[0,20]$ by $[-12,50] .$
(b) Approximately how much did the rat weigh after 7 days?
(c) Approximately when did the rat's weight reach 27 grams?
(d) Approximately how fast was the rat gaining weight after 4 days?
(e) Approximately when was the rat gaining weight at the rate of 2 grams per day?
(f) Approximately when was the rat gaining weight at the fastest rate?

Key Concepts

Differentiation

Differentiation is the process of finding the derivative of a function, which represents its instantaneous rate of change. In applied contexts, such as this experiment, the derivative provides crucial information about how quickly variables like weight are changing at any given moment.

Solving Equations

Solving equations involves finding the values of the variable that make the equation true. In the context of this experiment, it includes determining the time when the baby rat reaches a specific weight, which often requires numerical methods or algebraic manipulation, especially when dealing with polynomial equations.

Critical Points and Optimization

Critical points of a function are values of the variable where the derivative is zero or undefined, often corresponding to local maximums or minimums. Identifying these points is key in optimization problems, such as determining the time when the rate of weight gain is at its highest.

Graphing Functions

Graphing functions involves plotting the set of points that satisfy the function's equation on a coordinate system. This is crucial for visualizing the behavior of a function over a specified domain and range, identifying key features like intercepts, turning points, and intervals of increase or decrease.

Polynomial Functions

Polynomial functions are algebraic expressions involving sums of powers of an independent variable multiplied by coefficients. They are fundamental in modeling a wide range of real-world phenomena, and their behavior—as determined by the degree and the leading coefficient—helps in understanding growth patterns, curvature, and long?term behavior.

Function Evaluation

Function evaluation is the process of substituting a specific value of the independent variable into a function to determine the corresponding output. This technique is essential for estimating real-world quantities at given points, such as approximating a baby rat's weight after a certain number of days in a growth study.

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