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A rechargeable battery is plugged into a charger. The graph shows $C(t),$ the percentage of full capacity that the battery reaches as a function of time $t$ elapsed (in hours).(a) What is the meaning of the derivative $C^{\prime}(t) ?$(b) Sketch the graph of $C^{\prime}(t) .$ What does the graph tell you?

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(a) $C^{\prime \prime}(t)$ is the instantancous rate of change of percentage of full capacity with respect to elapsed time in hours.(b) The graph of $C^{\prime}(t)$ tells us that the rate of change of percentage of full capacity is decreasing and approaching 0

02:43

Daniel Jaimes

01:06

Carson Merrill

03:05

David Mccaslin

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 8

The Derivative as a Function

Limits

Derivatives

Catherine A.

October 23, 2020

This will help alot with my midterm

Sharieleen A.

October 26, 2020

Daniel J, thanks this was super helpful.

Oregon State University

Idaho State 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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All right here we have a graph, it shows a charge percentage charge on a battery. Soc F. T. Is a percentage charge. So that at full charge we have 1.0. So 100%. And basically at four hours we're at .6 and so on. And anyway, that's a rechargeable battery. And we basically want to figure out what is the meaning of Si Prima T. And remember that means the derivative of C. With respect to time. So right away we know units will be percent charged per hour. So those would be the units for our derivative. The meaning is the rate of change a percent charge. Uh The rate of change of percent charge with respect to time and our units are in percentage charge per hour. Okay, so that is our C. Prima T. If we want to do a quick graph, I will make one on the right. So let's go ahead and get some straight lines here. So we'll make the graph of our derivative whoops. Just to get a rough sketch of art. Whoa! It is, there we go rough sketch of our derivative. Okay, so this will be then Si Prima T. Which will be in percent charge per hour versus time in hours. And we're just gonna do a rough sketch, notice it starts off if we look at our slopes um quite positive slope here, but the slope keeps getting less steep until it becomes zero slope. So we end up with something like a high positive initial slope and then it will taper off until something like that. So as you can see by the end, we're close to 00 slope. So that's um, what we see. So we can basically say that the charging rate slows with time and so basically the percent charge decreases with time and approaches zero in the end. So anyway, hopefully that helped have a wonderful day.

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