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Let $P(t)$ be the percentage of Americans under the age of18 at time $t .$ The table gives values of this function in operator name census years from 1950 to 2000 .$$\begin{array}{|c|c|c|c|}\hline t & {P(t)} & {t} & {P(t)} \\ \hline 1950 & {31.1} & {1980} & {28.0} \\ \hline 1960 & {35.7} & {1990} & {25.7} \\ \hline 1970 & {34.0} & {2000} & {25.7} \\ \hline\end{array}$$(a) What is the meaning of $P^{\prime}(t) ?$ What are its units?(b) Construct a table of estimated values for $P^{\prime}(t) .$(c) Graph $P$ and $P^{\prime}$ .(d) How would it be possible to get more accurate valuesfor $P^{\prime}(t) ?$

(a) $P^{\prime}(t)$ is the rate of change of $\%$ of americans under 18Unit of $P^{\prime}(t)$ is $\% / yr$(b) See table(c) See graph(d) The accuracy may be increased by using shorter time intervals when measuring rates of change.

Calculus 1 / AB

Chapter 2

DERIVATIVES

Section 2

The Derivative as a Function

Derivatives

Differentiation

Applications of the Derivative

Campbell University

Harvey Mudd College

Baylor University

Lectures

03:09

In mathematics, precalculu…

31:55

In mathematics, a function…

02:34

The following table shows …

02:11

POPULATION The projec…

02:40

The following table lists …

01:25

PETS Let $d(t)$ be the num…

04:50

The table shows the percen…

04:14

0:00

The table gives the number…

04:09

The following table gives …

04:52

The function $$P(a)=0.015 …

03:10

The unemployment rate $ U(…

so important. Everyone tell what is the meaning off a prime of tea? So that is the rate that which percentage or percent off Americans under the Angel three Teen Changes and the unit off p prime of tea would be percent her year Now For Part B, we have to construct a table for the values off P Pride up p prime of teeth. So let's just right the values for pft first. And then we use those to calculate the values for P prime of tea. So we're starting with the year 1950 that we have 1960 1970 duty in 1990 and 2000. The values of plt commend our 31.1 35.7 34 points you up 28 points, you know, 25.7 and 25.7 now help prevent when a calculated three prime of tea. So, for example, we cannot actually find the value three prime of tea for 1950. What we're gonna do is people use the population in 1960 which is 35.5 minus depopulation in 1950 which is 31.1 and divide that by the difference in those years so 1960 minus 1950 And that kiss a 0.46 So using that same procedure, we will go ahead and calculate the other people prime plan use so they come out to be negative 0.17 Negatives. Your 0.6 is your negatives. Your 0.23 and zero. So, using a formula P prime off T is equal to p off T plus one minus actually steep US 10 minus P of tea and then divided by 10. So that's the general formula used to calculate totals of P prime values. Now, in the next part, it says to graph p off tea and then p prime of tea. So let's grab pft. So we start with the year 1950. Then we have 60 1970 1980 1990 then the year 2000. So this is our tea in years, and we can start with, uh, 10 20 30 and 40. So this is pft. So in 1950 it was 31.1 soul. You can mark a point here 1960 it was 35.7. So we could mark a point here. And then it was 34 in 19 seventies will be marking the point here in 1980 it was 28 source on the run here. And then in 1990 it was 25.7. And also in 2000 reports 25.7. So we can join these points through a small car and that gives us the graph for pft. Now, the next thing we want to do is draw the graph for four p prime of tea. So we start with prime of tea. So again we have 1950 that have 1960 1970 1980 1990 2000. So this is our tea and then p prime of tea. We can have negative one here, and we could have one here, So let's talk the value so we don't have any value for 1950 But for for 1960 we have a van off 19600.46 so we could mark a point here and then for 1970 we have negative 0.17 source of around here and then for 1980 we have negative 19800.6 around here and then is 0.23 So somewhere around here and then the last point we have is zero. So then we can use a smooth curve to join these points. And that's the graph for a P prime of tea. Now the next graph say. Next question says, How can you find a more accurate draft for pft? So, uh, Seymour accurate graph off, be prime of tea can be obtained is there are mawr. Bad news off pft, soem or values off pft will help us to generate a more accurate draft for peace.

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