Daylight function for 40^∘ N Verify that the function D(t)=2.8 sin((2 π)/(365)(t-81))+12 has the

Daylight function for 40^∘ N Verify that the function...

Key Concepts

Function Transformations

Function transformations involve systematically altering a basic function through scaling, translating, or reflecting its graph to better fit observed data or real-world scenarios. This process is fundamental in mathematical modeling as it provides the flexibility needed to tailor standard functions to describe a wide range of periodic behaviors accurately.

Maximum and Minimum Values

Maximum and minimum values are the highest and lowest points reached by a function during one complete cycle. These extreme values are key to understanding the range of oscillation in a model and are often associated with significant events or thresholds in real-world applications.

Vertical Shift

Vertical shift is the upward or downward translation of a function’s graph, determined by adding a constant term to the function. In the context of trigonometric models, it sets the central axis or baseline around which the oscillations occur, matching the average or equilibrium value observed in the data.

Period

The period of a function is the length of the interval after which the function's values start repeating. Determining the correct period is essential in modeling cyclical phenomena since it defines the cycle's duration and ensures that the model accurately reflects the repeating nature of the underlying process.

Amplitude

Amplitude refers to the magnitude of oscillation in a sine or cosine function, representing half the distance between the maximum and minimum values of the function. It indicates the strength or intensity of the periodic variation, playing a critical role in quantifying the extent of fluctuations in a modeled phenomenon.

Periodic Functions

Periodic functions are functions that repeat their values at regular intervals, known as periods. They are essential in modeling cyclical phenomena observed in nature, engineering, and various scientific fields. The concept of periodicity allows for the prediction and analysis of repetitive behaviors, such as seasonal variations, sound waves, or biological rhythms.

Phase Shift

Phase shift is the horizontal translation of a function’s graph along the x-axis, which changes the position of its peaks and troughs. This adjustment is crucial for aligning theoretical models with observed data by ensuring that key features of the cycle occur at the correct time.

Trigonometric Functions

Trigonometric functions, like the sine and cosine functions, naturally exhibit periodic behavior and are widely used to represent oscillatory phenomena. Their properties, including periodicity, make them ideal candidates for modeling processes that repeat over time. They also allow for adjustments through transformations which align the model with real-world data.

Transcript

00:01 From this equation of a sine function, we can determine what the parameters of the sign function are, figure out what the period is, the maximum and minima of this function, and other special values.

00:15 So first, we can relate our equation to the general sign function, which is sum a times the sign of b times t minus c, all plus d.

00:32 Where a is the amplitude, b is related to the period, c is the phase shift, and d is the vertical shift.

00:40 So from our equation, we know that a is the number out front, which is 2 .8.

00:48 B is the number in front of the time minus the phase shift, which in this case is 2 pi over 365.

00:58 C is the number subtracted from time, which is 81, and d is, is the last number added on after the sign, which will be 12.

01:12 And from this, we can figure out the period if we know b, since the period t is just 2 pi over the parameter b.

01:23 And we know b to be 2 pi over 365.

01:26 So we can plug that in.

01:28 And note that in the numerator, 2 pi is the same thing as 2 pi divided by 1.

01:33 So it's be 2 pi over 1 divided by 2 pi over 365.

01:39 And if you have two fractions being divided by each other, this is equivalent to the first fraction multiplied by the reciprocal of the bottom fraction, which would be 365 over 2 pi.

01:52 The factors of 2 pi are going to cancel each other out, so we'll just be left with a period of 365.

02:00 Next, we can look at the maxima and minima of this function, since we know that for any sine function, the sine function, the sine of theta, will also, always be less than 1 and greater than negative 1, as in the smallest value that sign can have is negative 1, and the largest value that sign can have is 1.

02:22 So if we use that in the equation that we have, the 2 .8 sign of 2 pi over 365, t minus 81 plus 12.

02:40 If the sign, since the sign in this equation, the sign is the only thing that will be changing with time.

02:48 If the sign is a minimum of negative 1, then this will be negative 2 .8 plus 12, which is just equal to 9 .2.

03:05 And if the sign is 1, then this will become 2 .8 plus 12, which is 14 .8.

03:17 So these are the maxima and minima of the sign function.

03:21 We can ask ourselves at what time do these values occur? and we can figure this out because we know that the sign of pi over 2 is equal to 1.

03:39 Therefore, if the sign at pi over 2 is equal to 1, then in this equation, the maximum has to occur when the inside of the sign function is equal to pi over 2.

03:53 So therefore, 2 pi over 365 times t minus 81 is equal to pi over 2 when the entire sign equation is equal to 14 .8.

04:09 So here we can solve for t by multiplying by 365 over 2 pi on both sides to get that t minus 81 is equal to 365 over 2 pi.

04:27 Times pi over two.

04:30 And now we can add 81 on both sides to get t alone...

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