The profits of a small company for each of the first five...

The profits of a small company for each of the first five years of its operation are given in the following table: $$\begin{array}{lc} \hline \text { Year } & \text { Profit in } \$ 1000 \mathrm{s} \\\hline 2010 & 6 \\2011 & 27 \\2012 & 62 \\2013 & 111 \\2014 & 174 \\\hline\end{array}$$ a. Plot points representing the profit as a function of year, and join them by as smooth a curve as you can. b. What is the average rate of increase of the profits between 2012 and $2014 ?$ c. Use your graph to estimate the rate at which the profits were changing in 2012

The profits of a small company for each of the first five years of its operation are given in the following table: $$\begin{array}{lc} \hline \text { Year } & \text { Profit in } \$ 1000 \mathrm{s} \\\hline 2010 & 6 \\2011 & 27 \\2012 & 62 \\2013 & 111 \\2014 & 174 \\\hline\end{array}$$
a. Plot points representing the profit as a function of year, and join them by as smooth a curve as you can.
b. What is the average rate of increase of the profits between 2012 and $2014 ?$
c. Use your graph to estimate the rate at which the profits were changing in 2012

Key Concepts

Graphical Estimation

Graphical estimation refers to the process of approximating numerical values, such as the rate of change or related measurements, by analyzing and interpreting the shape and characteristics of the plotted graph. This method is especially useful when exact calculations are challenging or when a visual assessment helps in understanding the function's behavior.

Instantaneous Rate of Change

The instantaneous rate of change represents the slope of the function at a specific point, which is equivalent to the derivative at that point. It describes how rapidly the function's value is changing at that particular moment.

Average Rate of Change

The average rate of change is determined by calculating the difference in the function's value between two points and dividing it by the difference in the independent variable over that interval. This measure provides an overall sense of how the function behaves between those points.

Graphing Functions

Graphing functions involves plotting ordered pairs to represent the relationship between an independent variable and its dependent variable on a coordinate plane. By connecting these points, one can visualize trends, patterns, and behaviors of the function over various intervals.

Transcript

00:01 We are given the profits of a small company and its first five years of operation from 2010 to 2014.

00:08 And then the profits are thousands of dollars.

00:10 So like that first year 2010, that's $6 ,000.

00:14 So i'm going to go ahead and plot these points to represent a graph and try and connect it in as smooth a curve as possible.

00:21 So 2010, i'm going to say, is the first full year.

00:24 And then that profit was $6 ,000.

00:26 So that's going to be way down here after that first full year, which is $20.

00:30 Just for reference there and $6 ,000, so like not even halfway to that $50 ,000.

00:38 And then 2011 is that next full year, so year two of business.

00:44 And it's $27 ,000, which should look like it's about halfway to $50 because obviously 25 would be halfway to 50.

00:51 Now year 3, i'm at $62 ,000, which has gone beyond that first mark here of 50.

00:58 And we'll put it somewhere just a little bit above 50.

01:01 And then for 2013, the next year over, i want to plot it at 111 ,000.

01:06 So i'm going to go above 100 to like 111.

01:11 And then for the last year, the fifth year of operation that we have here, is 174.

01:16 So that's going to be nearly halfway between 150 and 200.

01:20 So up here.

01:22 So if i tried to connect it with a smooth curve, it would look something probably like this.

01:33 And i'm not going to put any arrows on it because i don't really know what happens beyond.

01:37 And i can't obviously go to the left over here because time doesn't go into the negative.

01:42 So i don't have anything before the company started and i don't really know what's going to happen after.

01:47 So this is my graph for the information i'm given.

01:51 Now, i want to know what the average rate of increases between 2012 and 2014.

01:56 So for average rate, you're thinking average rate of change, which would typically be that f of b minus f of a setup all over b minus a.

02:08 So if we're trying to look from 2012 to 2014, well, on my graph, those aren't the years they're plotted.

02:17 Those are years three and years five because of the time we're saying this is the third year of operation and the fifth year of operation.

02:27 We didn't actually shift all the way over to the 2000s for this.

02:31 So i'm really just looking from 3 to 5, and i can find those points from my graph.

02:36 So if i fill an f of 5 minus f of 3 over 5 minus 3, f of 5 is the amount i had at that fifth point.

02:47 That's the 174.

02:50 Minus f of 3 is what i had at that third year.

02:53 That's 62 over 5 minus 3 is 2...

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