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When you turn on a hot-water faucet, the temperature $T$ of the water depends on how long the water has been running.$$\begin{array}{l}{\text { (a) Sketch a possible graph of } T \text { as a function of the time } t} \\ {\text { that has elapsed since the faucet was turned on. }} \\ {\text { (b) Describe how the rate of change of } T \text { with respect to } t} \\ {\text { varies as } t \text { increases. }} \\ {\text { (c) Sketch a graph of the derivative of } T .}\end{array}$$

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04:05

Dena Ho

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 8

The Derivative as a Function

Limits

Derivatives

Oregon State University

Harvey Mudd College

Baylor University

Boston College

Lectures

04:40

In mathematics, the limit of a function is the value that the function gets very close to as the input approaches some value. Thus, it is referred to as the function value or output value.

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.

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so looking at the graph, we want to determine what it looks like when we have a faucet that increases in temperature. We turn onto hot water. Um, and what happens? Well, we know that it starts off is cold. Um, it always starts off as cold, and then it quickly becomes warmer and warmer until it's the hottest temperature possible. Um, eso a decent representation of this would be this graph right here again, these mad. These numbers don't matter as much. You could do five or 10 whatever. But what we see is that we start off with a cold value. Um, the cold temperature if this y axis is temperature, and then as time goes on along the X axis, we see it's getting very hot quickly, and then it starts to stabilize. It breaches its maximum temperature because when we turn on a faucet, there's not a there's no way it could just keep heating up forever. There's there is a maximum point. So what that tells us, um, is Then if we take the derivative, we see what it looks like. Um, and this derivative is showing us if we look from zero to infinity that we have high temperature. Ah, very high rate of change. Rather, this derivative graph shows a high rate of change, a positive rate of change. So a quick increase in temperature. And then as time goes on, it's slower and slower. Um, as it increases in temperature, um, to the point where it becomes a minuscule amount of change.

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