Sleeping Habits The percentage of people (aged 15 and above)...

Sleeping Habits The percentage of people (aged 15 and above) in the United States who are asleep $t$ hours after 9: 00 P.M. can be modeled as $$ \begin{aligned} s(t) &=-2.63 t^{2}+29.52 t+13.52 \text { percent } \\ \text { data from } 0 & \leq t \leq 10 . \end{aligned} $$ (Source: Based on data from American Time Use Survey, March 2009, Bureau of Labor Statistics) a. Describe the direction and concavity of $s$ over $0 \leq t \leq 10$ b. Will a linearization of $s$ always overestimate the function values? Explain.

Sleeping Habits The percentage of people (aged 15 and above) in the United States who are asleep $t$ hours after 9: 00 P.M. can be modeled as
$$
\begin{aligned}
s(t) &=-2.63 t^{2}+29.52 t+13.52 \text { percent } \\
\text { data from } 0 & \leq t \leq 10 .
\end{aligned}
$$
(Source: Based on data from American Time Use Survey, March 2009, Bureau of Labor Statistics)
a. Describe the direction and concavity of $s$ over $0 \leq t \leq 10$
b. Will a linearization of $s$ always overestimate the function values? Explain.

Key Concepts

Quadratic Functions

A quadratic function is a polynomial of degree two represented in the form f(t) = at² + bt + c. The sign of the coefficient 'a' determines the direction in which the parabola opens; if 'a' is negative the function opens downward, and if positive it opens upward. This property is essential in analyzing the overall direction and behavior of the function over any interval.

Concavity

Concavity indicates how a function curves. A function is concave down if its second derivative is negative, meaning that the slope of the function is decreasing and the graph bends downward. Conversely, if the second derivative is positive, the function is concave up, with the graph bending upwards. Understanding concavity is important when predicting the behavior of functions and the quality of approximations.

Linearization (Tangent Line Approximation)

Linearization is a method used to approximate the value of a function near a given point by using the tangent line at that point. This approximation relies on the value and the first derivative of the function and provides a simpler linear model that is accurate for values of the independent variable close to the point of tangency.

Approximation Error in Linearization

The error in a linear approximation is closely tied to the curvature of the function, as described by its concavity. If a function is concave down, the tangent line typically overestimates the function values near the point of tangency, while for a concave up function, the tangent line might underestimate the values. This concept is crucial in evaluating when a linearization provides an overestimate or an underestimate of the actual function.

Transcript

Please give Ace some feedback