The rate at which the risk of Down syndrome is changing is...

The rate at which the risk of Down syndrome is changing is approximated by the function $r(x)=0.004641 x^{2}-0.3012 x+4.9 \quad(20 \leq x \leq 45)$ where $r(x)$ is measured in percentage of all births/year and $x$ is the maternal age at delivery. a. Find a function $f$ giving the risk as a percentage of all births when the maternal age at delivery is $x$ years, given that the risk of down syndrome at age 30 is $0.14 \%$ of all births. b. Based on this model, what is the risk of Down syndrome when the maternal age at delivery is 40 years? 45 years?

The rate at which the risk of Down syndrome is changing is approximated by the function
$r(x)=0.004641 x^{2}-0.3012 x+4.9 \quad(20 \leq x \leq 45)$
where $r(x)$ is measured in percentage of all births/year and $x$ is the maternal age at delivery.
a. Find a function $f$ giving the risk as a percentage of all births when the maternal age at delivery is $x$ years, given that the risk of down syndrome at age 30 is $0.14 \%$ of all births.
b. Based on this model, what is the risk of Down syndrome when the maternal age at delivery is 40 years? 45 years?

Key Concepts

Function Evaluation

Function evaluation involves substituting specific values into the derived function to compute the corresponding output. This step is fundamental in verifying the model and in making predictions or assessments based on the constructed function.

Initial Condition

An initial condition is a known value of the function at a specific point that is used to determine the constant of integration when recovering a function from its derivative. It is essential in ensuring that the solution to a differential equation or an integrated function is unique and fits the practical scenario being modeled.

Rate Function

A rate function represents the instantaneous rate of change of a quantity with respect to another variable. In mathematical modeling, it is often given as a derivative of an underlying function that accumulates the total quantity of interest. Understanding the rate function is crucial because it allows one to determine how the quantity evolves as the independent variable changes.

Antiderivative

The antiderivative is the process of reversing differentiation, commonly known as integration. It is used to recover the original function from its derivative. In many applied problems, such as modeling risks or accumulative phenomena, integrating the rate function yields a function that represents the total accumulated value over time or another variable.

Transcript

00:01 For this problem, we are told that the rate at which the risk of down syndrome is changing is approximated by the function r of x equals 0 .004641x squared minus 0 .3012x plus 4 .9, 4x between 20 and 45, where r of x is measured in percentage of all births per year and x is the maternal age at delivery.

00:23 For part a, we are asked to find a function, f, giving the risk as a percentage of all births when the maternal age at delivery is x years, given that the risk of down syndrome at age 30 is 0 .14 % of all births.

00:40 So, one moment here, f is just going to be the antiderivative of r of x there.

00:48 So we'd have 0 .004641 x cubed over 3, minus 0 .3012 x squared over 2, plus 4 .9, plus our initial value, we are told that, or actually it wouldn't be an initial value, we would have plus c, where we're told that f of 30 is going to be 0 .14, which i need to plug some numbers into a calculator here, one moment.

01:22 So plugging in x equals 30 into all the stuff in our f of x, that gives us 8 .7757, but of course we are still left with that c that we're trying to solve for.

01:35 So we'd get, one moment here, we'd get that c then must equal negative 8 .6357.

01:43 So we have f of x is going to equal.

01:46 I'm just going to copy and paste that stuff from above.

01:54 This minus this.

01:58 Uh -oh.

02:01 Appears the recording is continuing, but my tablet is not responding.

02:04 Hopefully, let's see here.

02:06 There we go...