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At 7 AM the outdoor temperature was $68^{\circ} \mathrm{F}$ and at $11 \mathrm{AM}$ the temperature was $76^{\circ} \mathrm{F}$, is there a time between 7 and 11 AM when the temperature was (a) $73^{\circ} \mathrm{F}$ ? (b) $71^{\circ} \mathrm{F}$ ? (c) $65^{\circ} \mathrm{F}$.

(a) yes(b) yes(b) possible but not guaranteed

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 3

Limits and Continuity

Derivatives

Oregon State University

Harvey Mudd College

University of Nottingham

Idaho State University

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

05:22

When is the temperature at…

02:22

At 8: 00 P.M., a temperatu…

02:03

The temperature at 6 A.M. …

02:01

The temperature $T,$ in de…

So in the given question, we have any question which has given us why is equal to minus eight X square plus 64 X minus 1 20 and we are told that X is always later than zero. And this expression is actually showing the data that when why has taken us the excess temperature, the access temperature about and though the normal body temperature a normal body temperature 98 point four degrees tonight which is measured in which is measured in terms of negative Maronite and the variable X right. So why is actually giving the data of the excess in temperature above the normal normal body temperature given in 98 point normal body temperature, which is 98.4 degrees Fahrenheit, which is measured in degrees Fahrenheit, and X, which is the time in hours? The time. And I was So this is what we have been given in the question and now we are asked that when the temperature is at its peak, when temperature yes, but it's big look at its peak. What is the value? What is the value of what is the what is the time and temporary check had that point. So this is what we need to fight. So when we are told that the temperature is at its peak, what is the time at which the temperature is at its peak? And what is the maximum temperature at that point? So, in order to find this, what we can do is we contain the equation. Why is equal to minus eight times minus eight minus eight times X squared plus 64 X minus 1 20. And what we can do over here first, as we can take minus eight as a common factor. And then what we would have is X square minus eight times X minus 1 20. Divided by eight is 16 divided by four. Or we can say this 15 right, eight times 15 is would as 1 20. So what we would have is plus 15. So this is what we would have. And now what we can do is we can write this equation as X minus for the whole square, minus one. And if we write it like this, we can see that the peak value of why would be obtained when X is equal to fall right when X is equal to four. The term inside the term in the expression. Why? Where? X minus. Where is there would become zero? And why would be I did speak so Speak value or bye. Yes, when X is equal to four. And this means that maybe such stood X equal to four. The value of what it would be equal to the value of y would be equal to minus eight times minus one, which is equal to eight. That is, it is eight units the temperature. Yes. Eight units, eight units. About 98.4 degrees Fahrenheit, which is equal to 106.45 tonight. 106 0.4 per night and four hours time. Right. Access equal to four. Which means the time is four hours. So this is the required answer. I hope you understood the method. This is the required answer. I hope you understood the metric Thank you.

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