Temperatures in Fairbanks, Alaska The graph in the...

Temperatures in Fairbanks, Alaska The graph in the accompanying figure shows the average Fahrenheit temperature in Fairbanks, Alaska, during a typical 365 -day year. The equation that approximates the temperature on day $x$ is $$y=37 \sin \left[\frac{2 \pi}{365}(x-101)\right]+25$$ and is graphed in the accompanying figure. a. On what day is the temperature increasing the fastest? b. About how many degrees per day is the temperature increasing when it is increasing at its fastest? (GRAPH CAN'T COPY)

Temperatures in Fairbanks, Alaska The graph in the accompanying figure shows the average Fahrenheit temperature in Fairbanks, Alaska, during a typical 365 -day year. The equation that approximates the temperature on day $x$ is
$$y=37 \sin \left[\frac{2 \pi}{365}(x-101)\right]+25$$
and is graphed in the accompanying figure.
a. On what day is the temperature increasing the fastest?
b. About how many degrees per day is the temperature increasing when it is increasing at its fastest?
(GRAPH CAN'T COPY)

Key Concepts

Optimization Using Derivatives

Optimization in calculus involves finding the points at which a function reaches its maximum or minimum values by analyzing its derivatives. In the context of a rate of change or derivative function, determining when this derivative is maximized helps identify the point at which the original function is increasing most rapidly. This technique is widely used across various applications, from physics to economics, to improve performance or understand peak behavior in dynamic systems.

Derivatives

Derivatives in calculus represent the instantaneous rate of change of a function with respect to its independent variable. When applied to a function modeling a physical process, the derivative indicates how quickly the process is changing at any given point. This concept is fundamental for understanding concepts such as velocity in motion or the rate at which temperature is changing over time.

Periodic Functions

Periodic functions are functions that repeat their values in regular intervals or periods. They are commonly used to model cyclical phenomena such as seasonal temperature variations. Understanding the periodic nature of a function allows one to predict behavior over time and analyze patterns that recur at consistent intervals.

Trigonometric Function Properties

Trigonometric functions, such as the sine function, have inherent properties including amplitude (which measures the peak deviation from the average value), period (which determines the length of one complete cycle), phase shift (which horizontally translates the graph), and vertical shift (which moves the graph up or down). These properties are crucial in accurately modeling and interpreting real-world data that vary in a regular cyclical manner.

Transcript

00:01 Section 3 .6, problem number 100.

00:05 So here we're dealing with a periodic function, a sign function that models the temperature from fairbanks, alaska.

00:15 So they're saying that, and they draw you a curve here, they're saying this particular sign function here is a cyclical model that shows the temperature in degrees fahrenheit for every day and a 365 -day year for fairbanks, alaska.

00:31 In part a, they ask us to figure out, so in part a, they want to know on what day is the temperature increasing the fastest.

00:45 So what day is the temperature increasing the fastest? so when i look at rates of increase or decrease, i'm really talking about the first derivative here.

01:07 So what is the derivative of this function? so let's differentiate this and come up with a solution.

01:12 So you know that y prime is going to be equal to.

01:17 So this is 37.

01:19 What is the derivative of the sine function? well, that's the cosine function.

01:24 So 2 pi over 365 times x minus 10.

01:33 And then 25, the derivative that is 0.

01:36 Now with the chain rule, what's the derivative of what you see inside those parentheses? that derivative is 2 pi over 3 .7.

01:45 So what it tells me is that if you look at my derivative y prime is going to be 2 times 37 is what 74 so this is going to be 74 pi over 365 times a cosine 2 pi over 365 times a cosine 2 pi over 365 x minus 101 so that is my derivative function.

02:25 So when will this be increasing the fastest? so you know that the cosine portion of this is a number between negative 1 and 1.

02:34 So the maximum it could be would be 1...

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