AP Precalculus Practice Exam Question 84

NumWorks
23 May 202303:11
EducationalLearning
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TLDRThe speaker in the video script discusses determining the number of completed cycles of the graph of a given function f(x) = 2sin(4x) + cos(2x) on the interval from 0 to 1000. They use a grapher application to find the period of the function by identifying two maximum points and calculating the difference in their x-values. The period is found to be approximately ฯ€, and by dividing 1000 by this period, they conclude that there are about 318 full cycles within the given interval.

Takeaways
  • ๐Ÿ“š The task involves analyzing the function f(x) = 2sin(4x) + cos(2x) to determine the number of completed cycles on the interval from 0 to 1000.
  • ๐Ÿ” The script describes using a grapher application to visually determine the period of the function.
  • ๐Ÿ“ˆ The period is calculated by finding the difference between two maximum points on the graph of the function.
  • ๐Ÿ“ The first maximum point identified in the script is at x = 0.352.
  • ๐Ÿ”ข The second maximum point is at x = 3.494, and the difference between these points is used to estimate the period.
  • ๐Ÿงฎ The period is approximately calculated to be 3.142, which is close to the mathematical constant ฯ€.
  • โž— To find the number of completed cycles, the interval length (1000) is divided by the estimated period.
  • ๐Ÿ”‘ The result of this division is approximately 318, which represents the number of full cycles within the given interval.
  • ๐Ÿค” The script acknowledges that the period is likely exactly ฯ€, but the calculation uses the rounded value for the estimation.
  • ๐Ÿ“‰ The process involves a combination of graphical analysis and arithmetic calculation to solve the problem.
  • ๐Ÿ“ The final answer is presented as 318 full cycles completed on the interval from 0 to 1000 for the function f(x).
Q & A

  • What is the given function f(x) in the script?

    -The given function is f(x) = 2 * sin(4x) + cos(2x).

  • What is the task described in the script?

    -The task is to determine the number of completed cycles of the graph of function f on the interval from 0 to 1000.

  • What is the approach to find the period of the function f(x)?

    -The approach is to use a grapher application to plot the function and identify the period by finding the difference between two consecutive maximum points (Max - Max).

  • How does the script suggest to calculate the period of the function?

    -The script suggests calculating the period by identifying two maximum points on the graph and subtracting their x-values.

  • What are the two maximum points identified in the script for calculating the period?

    -The two maximum points identified are x = 0.352 and x = 3.494.

  • What is the calculated period of the function f(x) according to the script?

    -The calculated period is approximately 3.142, which is likely to be ฯ€ (pi).

  • How is the number of completed cycles determined in the interval from 0 to 1000?

    -The number of completed cycles is determined by dividing the length of the interval (1000) by the period of the function.

  • What is the final answer for the number of completed cycles in the interval from 0 to 1000?

    -The final answer is approximately 318 full cycles.

  • Why is the period likely to be ฯ€ (pi) in the script?

    -The period is likely to be ฯ€ because the sine and cosine functions have a fundamental period of 2ฯ€, and the given function has a combination of these with coefficients that could simplify to a period of ฯ€.

  • What is the significance of the period in analyzing the graph of a function?

    -The period signifies the length of one complete cycle of the function's graph, which is crucial for understanding the function's behavior and predicting its values over different intervals.

  • How can the understanding of the function's period be applied in real-world scenarios?

    -The understanding of a function's period can be applied in various fields such as physics for wave analysis, engineering for signal processing, and economics for analyzing cyclical trends.

Outlines
00:00
๐Ÿ“š Calculating the Number of Completed Cycles

The speaker is addressing a math problem involving the function f(x) = 2sin(4x) + cos(2x). They aim to find the number of completed cycles of the graph of f on the interval from 0 to 1000. The process involves determining the period of the function using a graphing application, identifying two maximum points, and calculating the period by subtracting their x-values. The speaker finds the period to be approximately 3.142, which they suggest is likely equal to ฯ€. They then calculate the number of full cycles by dividing 1000 by the period, arriving at approximately 318 complete cycles.

Mindmap
Keywords
๐Ÿ’กFunction f
In the context of the video, 'Function f' refers to a mathematical function given by the expression 'f(x) = 2 * sin(4x) + cos(2x)'. This function is the central theme of the video, as the analysis revolves around determining the properties of this specific function. The function combines sine and cosine components, which are trigonometric functions, to create a complex wave pattern.
๐Ÿ’กPeriod
The 'Period' of a function is the length of one complete cycle of the function's graph. In the video, determining the period of function f is crucial because it helps to understand how often the function repeats its values. The period is calculated by identifying two maximum points on the graph and subtracting their x-values, which in the script is approximated to be ฯ€.
๐Ÿ’กSine
'sine' is a trigonometric function represented as 'sin(x)', which describes the ratio of the length of the side opposite to an angle in a right-angled triangle to the length of the hypotenuse. In the video, 'sin(4x)' is part of the function f, indicating that the sine function is scaled and shifted in the x-direction.
๐Ÿ’กCosine
'Cosine' is another trigonometric function, symbolized as 'cos(x)', which relates to the adjacent side of an angle in a right-angled triangle compared to the hypotenuse. In the script, 'cos(2x)' is a component of function f, showing a different frequency than the sine component.
๐Ÿ’กGraph
A 'Graph' in the mathematical context refers to the visual representation of a function, where the x-axis represents the input values and the y-axis represents the output values. The video discusses plotting the graph of function f to visually determine its properties, such as its period.
๐Ÿ’กCycle
A 'Cycle' in the context of the video refers to a complete repetition of the function's pattern. The number of cycles completed on the interval from 0 to 1000 is what the video aims to calculate, which is directly related to the period of the function.
๐Ÿ’กInterval
The 'Interval' in the video refers to the range of x-values being considered, which is from 0 to 1000. This interval is important because it sets the bounds within which the number of completed cycles is to be determined.
๐Ÿ’กGrapher Application
A 'Grapher Application' is a tool used to plot and analyze the graph of mathematical functions. In the script, it is mentioned as the means to visualize function f and identify its period by finding maximum points.
๐Ÿ’กMaximum Points
In the context of the video, 'Maximum Points' refer to the highest points on the graph of the function. These points are used to calculate the period by subtracting the x-values of two such points, which provides an estimate of the function's period.
๐Ÿ’กCalculation App
A 'Calculation App' is a software tool used for performing mathematical calculations. In the script, it is used to subtract the x-values of two maximum points to find the period of the function.
๐Ÿ’กTrigonometric Functions
Trigonometric Functions, such as sine and cosine, are mathematical functions that relate the angles of a triangle to the lengths of its sides. In the video, these functions are integral to the definition of function f and understanding its behavior.
Highlights

The function f(x) is given as f(x) = 2sin(4x) + cos(2x).

The task is to determine the number of completed cycles of the graph of f in the XY plane on the interval from 0 to 1000.

The period of f is crucial for calculating the number of completed cycles.

The period is defined as the length of one cycle of the function.

A grapher application will be used to determine the period of the function.

The graph of the function is plotted to visualize the curve.

Identifying two maximum points on the graph to calculate the period.

The first maximum point is found at x = 0.352.

The second maximum point is identified at x = 3.494.

The period is calculated by subtracting the x-values of two maximum points.

The calculated period is approximately 3.142, likely equal to ฯ€.

The number of completed cycles is found by dividing the interval length (1000) by the period.

The result of the calculation is approximately 318 full cycles.

The use of a calculator app for the period calculation is mentioned.

The transcript provides a step-by-step approach to solving the problem.

The solution process involves both graphical and numerical methods.

The transcript emphasizes the practical application of mathematical concepts.

Transcripts

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TrigonometryFunction AnalysisPeriod CalculationGraph PlottingCycle CountMax PointsInterval AnalysisMathematicsEducational ContentSine Cosine