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The number, $N$ (in million) of VCR's sold in the United States for the years 1990 to 2001 may be approximated by the formula $N=(12 t+1) /(t+2)$, where $t$ is the time in years, with 1990 being $t=0 .$ (a) Find the number sold in $1990,1994,$ and $2000 .$ (b) Find the rate of change of sales in 1990,1994, and 2000.

(a) $1 / 2,49 / 6,121 / 12$ million(b) $92 / 25,828 / 3721,3312 / 21025$ million per year

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 10

Related Rates

Derivatives

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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.

04:45

(Continuation of previous …

07:56

Sales The rate of change i…

05:36

Sales The sales of a new p…

03:19

The sales of a new persona…

00:38

Sales Often sales of a new…

01:07

Electronics Sales (1990s) …

03:34

The sales $S$ (in millions…

03:42

Find the time $t$ in years…

02:24

for this problem. We've been given an equation that shows the number of VCR sold in the United States over a certain period of time. And his N number of VCRs equals 12 t plus one over T plus two. Now, T is the number of years after 1990. So there's three different years that we're going to be looking at here. 1990 1994 2000. So for 1990 were calling that to equaling zero. So four years later, TS four for 1994 and she is 10 for 2000. So we'll just be referring to that some because we are gonna be looking at those three years a couple of different times. Okay, First, let's use our formula here and actually find the number of VCR sold in each of these years. So 1990 when t is zero 1994 when t is four in 2000, when she is 10. Okay, t equaling zero first. If I plug that in, well, that's just going to give me a one in the numerator and a two in the denominator. And I'm not gonna write this for every single one But remember, our units are million's of VCR sold. So it's not just half a VCR. It's half a million sold. Okay, what if ts four Well, that gives me 48 plus 1. 49 in the numerator and six in the denominator. So 49 6 million units. And if t is 10, that gives me 121 over 12. Okay, so that's just using our formula. Now, let's take the derivative because our second part of this problem is the rate of change of sales. So we're looking at D n d t. How is the sales changing with respect to time. So we have a quotient we need to use the quotient rule. It's the denominator times the derivative of the numerator, minus the numerator times the derivative of the denominator, which is just one all over the denominator squared. Okay, so let's take a look at this at three different years again. 1990 1994 in 2000. So what is d n d t at thes two points, right. And I will say I'm just leaving this d n d t. I could straighten that up and clean it up a little bit just because I'm just gonna be plugging in some points here. I'm just gonna leave it as is. If you want to make that look a little nicer with some algebra, you are more than welcome to. Okay, so 1990 t is zero. So that's going to give me 24 minus one over four or 23 4th. Okay, 1994. Now, T is four. That's going to give me six times 12 or 72. It's four that's going to be minus 49 and my denominator is gonna be six squared or 36. So that simplifies to 23 36 and for t equaling 10. That's going to give me 100. What? Let me just write that here. This d n d t keep a consistent. Okay, so again, tea is 10. That's 12 times 12 144 t is 10. That's going t minus 121 over 12 squared, which is 144 which simplifies to 23 over 144

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