AP Precalculus: Session 4

Modeling Views and Time Scale Conversions

Scenario Context: The number of views for an online musical performance is modeled by the function $T(t) = 10 \times 1.6^t$ . - $T(t)$ represents views in hundreds. - $t$ represents the time in months after the live performance.
Objective: Convert the function to represent the number of views in terms of $s$ days recorded after the live performance.
Assumptions for Conversion: - Each month consists of exactly $30$ days. - The relationship between days ( $s$ ) and months ( $t$ ) is defined as $s = 30t$ .
Step-by-Step Substitution: - To isolate $t$ , solve the relationship equation: $t = \frac{s}{30}$ . - Substitute the expression for $t$ into the original function: $v(s) = 10 \times 1.6^{\frac{s}{30}}$ .
Equivalent Expressions: - Using the laws of exponents, the expression can be rewritten to show the internal structure of the exponent: $10 \times 1.6^{(\frac{1}{30}) \times s}$ . - This matches Answer Choice C in the provided assessment material.

Technological Proficiency in AP Precalculus

General Expectation: Students are expected to use technology and calculators for specific designated portions of the AP Precalculus exam. Manual calculation is not always sufficient or required for all problems.
Required Computational Skills: - Graphing and Analysis: The ability to graph functions and interpret the resulting visual data. - Table Generation: Generating a table of values for a specific function to analyze discrete data points. - Zero Finding: Locating the real zeros (x-intercepts) of a function. - Intersection and Extremas: Finding points of intersection between multiple graphs, as well as identifying the relative maximum and relative minimum values of a function. - Numerical Solving: Solving equations involving one variable numerically. - Regression and Residuals: Using technology to find a regression equation to model data and subsequently plotting the corresponding residuals to check the model's fit.

Analyzing Function Intersections Using Technology

Problem Description: Consider two functions, $f(x)$ and $g(x)$ , to find input values where output values are identical. - Function $f(x) = 1 + \log_{10}(x^4)$ . - Note on Notation: Binary log base $10$ is the common log. Therefore, $1 + \log_{10}(x^4)$ is equivalent to $1 + \log(x^4)$ . - Function $g(x) = \ln(6x + 2)$ .
Goal: Identify all input values ( $x$ ) where $f(x) = g(x)$ .
Methodology: Use a graphing utility to plot both functions and identify their points of intersection.
Results from Technology: - Intersection Point 1: Input value $x = -0.283$ . The shared output value at this point is $-1.194$ . - Intersection Point 2: Input value $x = 3.331$ . The shared output value at this point is $3.091$ .
Conclusion: The correct input values resulting in the same output are $x = -0.283$ and $x = 3.331$ (Answer choice C).

Modeling Theme Park Attendance via Logarithmic Functions

Scenario Context: The number of guests entering a theme park over time ( $t$ hours since opening) is modeled by the function $g(t)$ . - Function: $g(t) = 3.52 \times \ln(t + 0.82) + 0.75$ . - Unit Measurement: $g(t)$ is measured in thousands of guests.
Goal: Determine the time $t$ at which $7,500$ guests have entered the park.
Constraint Management: Since the model measures in thousands, the value $7,500$ must be converted. The equation to solve is $g(t) = 7.5$ .
Technological Application: Use the calculator's numerical solver to find the solution for: - $7.5 = 3.52 \times \ln(t + 0.82) + 0.75$
Result: The calculated time is $t = 5.985$ hours (Answer choice B).

Cubic Regression and Velocity Analysis

Scenario Context: A particle moves along a horizontal line for time $t \geq 0$ seconds. Velocity $v(t)$ is measured in feet per second.
Data Set: - $t = 0, v(t) = -35$ - $t = 1, v(t) = 10$ - $t = 2, v(t) = 5$ - $t = 3, v(t) = -20$ - $t = 4, v(t) = -40$ - $t = 5$ (no specific value recorded in the dialogue for this point, but used as an input boundary).
Regression Modeling: - Perform a cubic regression to find a model of the form $v(t) = at^3 + bt^2 + ct + d$ . - Residuals: Observe that the regression line does not pass perfectly through every point; the gap between the point and the line is the residual/error.
Strategy for Accuracy: - Store the variables $a, b, c,$ and $d$ in the calculator's memory or store the entire regression function in the function list for further calculation. Do not use rounded values mid-calculation.
Goal: Find when the velocity first changes from decreasing to increasing based on the regression model.
Calculus/Logic Connection: A change from decreasing to increasing corresponds to a relative minimum.
Results: - Using the calculated regression, the technology identifies the relative minimum at $t = 6.633$ . - At this time ( $t = 6.633$ ), the velocity output is $-60.169$ .

Polar Coordinate Systems and Average Rate of Change

Scenario Context: A car's position is modeled in a polar coordinate system starting at the origin. - Polar function: $r = f(\theta) = 2\theta - \sin(\theta)$ . - Interval: $0 \leq \theta \leq \pi$ .
Concept: Average Rate of Change (AROC): - Defined as the two-point slope: $\frac{f(b) - f(a)}{b - a}$ . - Applied to the interval $[0, \pi]$ : - $r(\pi) = 2\pi - \sin(\pi) = 2\pi - 0 = 2\pi$ . - $r(0) = 2(0) - \sin(0) = 0 - 0 = 0$ . - $AROC = \frac{2\pi - 0}{\pi - 0} = 2$ .
Interpretation: This rate implies that for every increase of $1$ radian in $\theta$ , the radial distance $r$ increases by approximately $2$ units.
Goal: Estimate the distance from the origin when $\theta = \frac{\pi}{4}$ . - Estimation Method: $Starting \, Distance + (Rate \, of \, Change \times Change \, in \, \theta)$ . - Estimation calculation: $0 + (2 \times \frac{\pi}{4}) = \frac{\pi}{2}$ .
Result: Convert $\frac{\pi}{2}$ to a decimal: $1.571$ (Answer choice B).

Questions & Discussion

Becky and Jamil Dialogue: - Becky: Initiates the view conversion problem and explains the substitution of $t = \frac{s}{30}$ . - Jamil: Introduces the technology skills required for the AP Precalculus exam, emphasizing numerical solving and regressions. - Becky: Demonstrates how to use a graphing utility to find intersections for the logarithmic functions $f(x)$ and $g(x)$ . - Becky: Walks through the transition of $7,500$ guests into the decimal $7.5$ to match the model's scale in thousands. - Jamil: Explains the cubic regression process and the importance of using stored values for accuracy. He highlights the definition of a relative minimum as the point where a function switches from decreasing to increasing. - Jamil: Concludes with the polar coordinate problem and the definition of Average Rate of Change as a tool for distance estimation.

Summary of Essential Terms

Intersection: The location(s) where the graphs of two functions cross; at these points, the functions have equal input and output values.
Relative Minimum (Local Minimum): The point at which a function's behavior changes from decreasing to increasing.
Relative Maximum (Local Maximum): The point at which a function's behavior changes from increasing to decreasing.
Average Rate of Change (AROC): A measure of the average change of a function over an interval, calculated via the slope formula: $\frac{f(b) - f(a)}{b - a}$ .

The function modeling the number of views for an online performance is given by $T(t) = 10 imes 1.6^t$ , with $t$ representing time in months and $T(t)$ in hundreds of views.
Converting months to days involves the relationship $s = 30t$ , leading to the function $v(s) = 10 imes 1.6^{ rac{s}{30}}$ , illustrating how to interpret data in terms of different time scales.
Technology plays a crucial role in AP Precalculus, enabling students to perform graph analysis, regression modeling, and numerical solving rather than relying solely on manual calculations.
Identifying points of intersection between functions, such as $f(x)$ and $g(x)$ , can be efficiently achieved using graphing utilities, showcasing the application of technology in mathematical analysis.
Average Rate of Change (AROC) is a valuable concept for analyzing function behavior and estimating values based on changing variables, particularly within polar coordinate systems.