Because of medical technology advances, the disability rates...

Because of medical technology advances, the disability rates for people over $65 \mathrm{yr}$ old have been dropping rather dramatically. The function $$ R(t)=26.3 e^{-0.016} \quad(0 \leq t \leq 18) $$ gives the disability rate $R(t)$, in percent, for people over age 65 from $1982(t=0)$ through 2000 , where $t$ is measured in years. a. What was the disability rate in $1982 ?$ In $1986 ?$ In 1994 ? In 2000 ? b. Sketch the graph of $R$.

Because of medical technology advances, the disability rates for people over $65 \mathrm{yr}$ old have been dropping rather dramatically. The function
$$
R(t)=26.3 e^{-0.016} \quad(0 \leq t \leq 18)
$$
gives the disability rate $R(t)$, in percent, for people over age 65 from $1982(t=0)$ through 2000 , where $t$ is measured in years.
a. What was the disability rate in $1982 ?$ In $1986 ?$ In 1994 ? In 2000 ?
b. Sketch the graph of $R$.

Key Concepts

Time Domain

The time domain specifies the interval over which the function is defined. It is important for understanding the range of time values for which the model is valid, ensuring that any analysis or graphing of the function is done within the appropriate context.

Decay Constant

The decay constant in an exponential model is the coefficient in the exponent that determines the rate at which the function decreases. A larger decay constant means a faster decrease, and it plays a critical role in quantifying how quickly improvements or changes occur over time.

Graphing Exponential Functions

Graphing an exponential decay function involves plotting a smooth, continuously decreasing curve that starts from the initial value and approaches, but never actually reaches, zero. Critical aspects include accurately marking the initial point, understanding the curvature, and interpreting how the function behaves over the specified time period.

Exponential Decay Function

An exponential decay function models processes where a quantity decreases at a rate proportional to its current value. This type of function is common in situations where the change over time is not linear, such as in population studies, radioactive decay, and improvements in health-related rates, where the decay rate is determined by the exponential term in the function.

Initial Value

The initial value in an exponential function represents the value of the function at the starting time (t=0). It provides a baseline measure from which all future values can be compared and is crucial for understanding the magnitude of the process at its beginning.

Transcript

00:01 Since the creation of the medicare and the medicaid programs in 1965, the percent of persons 65 years and over with family income below the poverty level has declined.

00:18 The percent can be approximated by the function bft equals 30 .60 minus 5 .79 times the natural logarithm of t, where t represents the number of years since 1965.

00:44 So we are going to find the value of the person p of t and its derivative in the years in 1970 in part a, 1990 in part b, and 2010 in par c.

01:04 Then in part d, we're going to talk about what happens.

01:08 To the rate of change over time.

01:11 Knowing that the rate of change of the percent of persons 65 years and over with family income below the poverty level is the derivative of p respect to t.

01:27 So first in part a before doing the calculations we are going to find the derivative of the function.

01:49 And we know that the derivative of this function, p of t is equal to this derivative of this constant is zero and the derivative of this term here is negative 5 .79 times the derivative with respect to t of the natural logarithm of t this is equal to negative 5 .79 times 1 over t that is negative 5 .79 over t so given this result we can now proceed to calculate both the value of the function p of t and its derivative p derivative of t in the years 1917 1990 and 2010 so we got to remember that t is the number of years since 1965 so when we want to calculate both p of t and its derivative at in the year 1970 we got to calculate how many years passed since 1965 so in part a we can say that t is equal to five because there this is a number of years since 1965 when we are in the year 1970.

03:40 You can say that this is true since 1970 minus 1965 is equal to 5.

03:54 And so the percent of of persons 60 years and over with family income below poverty level is p of 5, this case, this is equal to 30 .60 minus 5 .79 times natural logarithm of 5.

04:24 And now this value is approximately equal to 21 .28...

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