Social Security Assets The projected year-end assets in the...

Social Security Assets The projected year-end assets in the Social Security trust funds, in trillions of dollars, where $t$ represents the number of years since 2000 , can be approximated by $$A(t)=0.0000329 t^{3}-0.00450 t^{2}+0.0613 t+2.34$$ where $0 \leq t \leq 50 .$ Source: Social Security Administration. a. Where is $A(t)$ increasing? b. Where is $A(t)$ decreasing?

Social Security Assets The projected year-end assets in the Social Security trust funds, in trillions of dollars, where $t$ represents the number of years since 2000 , can be approximated by
$$A(t)=0.0000329 t^{3}-0.00450 t^{2}+0.0613 t+2.34$$
where $0 \leq t \leq 50 .$ Source: Social Security Administration.
a. Where is $A(t)$ increasing?
b. Where is $A(t)$ decreasing?

Key Concepts

Derivatives

Derivatives measure the instantaneous rate of change of a function with respect to its variable. In the context of this problem, the derivative of the asset function is used to determine whether the function is increasing or decreasing over a given interval.

Increasing Functions

An increasing function is one in which the output values rise as the input increases. When the derivative of a function is positive over an interval, it indicates that the function is increasing on that interval.

Decreasing Functions

A decreasing function is one where the output values fall as the input increases. This behavior is identified when the derivative is negative over a certain interval, showing that the function is declining in that range.

Critical Points

Critical points occur where the derivative is zero or undefined. In analyzing the behavior of a function, these points are crucial since they often mark the boundaries between intervals where the function switches from increasing to decreasing, or vice versa.

Transcript

00:01 For this problem, we have been given an equation a of t that shows the assets in a social security trust fund at year t, where t is the number of years past 2000.

00:12 So if t is 1, we're in 2001.

00:15 If t is 5, we're in 2005, and so on.

00:18 In order to find where this function is increasing and decreasing, our first step is to find the derivative.

00:24 So the derivative of our function a is going to be 0 .0 .0 .0.

00:31 0 .00987t squared minus 0 .009t plus 0 .0613.

00:44 So i'm just going to put a little box here so we don't lose this.

00:49 So that's the derivative of my function a.

00:52 Now, we need to see where this derivative is equal to 0 because that will tell us at what points my function could possibly be changing direction from, increasing to decreasing.

01:05 So we're going to set this equal to zero and solve.

01:09 Well, i'm certainly not going to try to use factoring to find my zeros.

01:13 So i'm going to go right to the quadratic formula.

01:18 So in this case, here is my a, here is my b, and here is my c.

01:25 So i get negative b, so the opposite of b, plus or minus the square root of b squared, minus, minus, 4 times a times c, all over 2a.

01:54 Okay.

01:56 Plugging this into a calculator, i get two different values for t.

02:00 I get either 83 .8 or i get a value of t equal to 7 .4.

02:10 Now, we are told that we only want to look at values of t between 0 and 50.

02:16 So that means that while that is in point of interest in this equation, we are not going to be looking at that.

02:21 It's outside of our range.

02:23 So we have one point where i could be possibly changing direction, 7 .4, which is about the year 2007...

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