Social Security Assets As seen in the first section of this...

Social Security Assets As seen in the first section of this chapter, 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 $A(t)=0.000039 t^{3}-0.00450 t^{2}+0.0613 t+2.34$ where $0 \leq t \leq 50$ . Find the value of $t$ when Social Security assets will decrease most rapidly. Approximately when does this occur? Source: Social Security Administration.

Social Security Assets As seen in the first section of this chapter, 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
$A(t)=0.000039 t^{3}-0.00450 t^{2}+0.0613 t+2.34$
where $0 \leq t \leq 50$ . Find the value of $t$ when Social Security assets will decrease most rapidly. Approximately when does this occur? Source: Social Security Administration.

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Key Concepts

Critical Points in Optimization

Critical points occur where the first derivative of a function equals zero or is undefined, and they are key in finding local extrema, such as maximum or minimum rates of change. In optimization problems, identifying critical points helps determine moments when a function's behavior, such as the rate of decline, is most extreme.

Second Derivative Test

The second derivative of a function provides information about the function's curvature or concavity. By examining the second derivative at a critical point, one can determine whether the function has a local maximum or minimum there. A positive second derivative implies concave up (a local minimum of the derivative), and a negative second derivative implies concave down (a local maximum of the derivative), which is essential for determining when a quantity decreases most rapidly.

Derivative as Rate of Change

The derivative of a function measures the rate at which the function's value changes with respect to changes in its input. In applied contexts like economic models or asset evaluations, the derivative indicates how quickly a quantity (such as assets) is increasing or decreasing at any point in time.

Transcript

00:01 All right, here we have a function.

00:04 This is the assets of social security and trillions of dollars, where t is the time in years after the year 2000.

00:13 We're going to carefully write this down because there are lots of zeros involved.

00:32 There got to be careful with zeros when you're dealing with money.

00:49 Okay.

00:53 All right, and so we want to find...

00:56 Okay, so we want to find...

00:57 So we also have a domain restriction.

00:59 0 less than t, less than 50.

01:07 Okay, so we want to find the t value or the year, the approximate year, when a of t decreases most rapidly.

01:16 Now the key to understanding this question is to realize that what you're looking for is the absolute minimum of not a of t.

01:29 You want to know when it's decreasing.

01:31 Increasing most rapidly.

01:35 So we want the absolute minimum of a prime of t.

01:41 So when is that? well, first of all, let's actually write down a prime of t, the function we actually want to minimize.

01:53 And here we go.

01:57 Let's compute some of these derivatives and i actually want to use a calculator here.

02:05 Okay, so this first derivative, 0 .0000, and basically i'm just multiplying 39 times 3, so that's 117 t squared minus this one's not so bad, 0 .09t plus 0 .0613...

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