Pupil size When the brightness x of a light source is...

Pupil size When the brightness $x$ of a light source is increased, the eye reacts by decreasing the radius $R$ of the pupil. The dependence of $R$ on $x$ is given by the function $R(x)=\sqrt{\frac{13+7 x^{0.4}}{1+4 x^{0.4}}} $ where $R$ is measured in millimeters and $x$ is measured in appropriate units of brightness. (a) Find $R(1), R(10),$ and $R(100) .$ (b) Make a table of values of $R(x)$ (c) Find the net change in the radius $R$ as $x$ changes from 10 to 100 .

Pupil size When the brightness $x$ of a light source is increased, the eye reacts by decreasing the radius $R$ of the pupil. The dependence of $R$ on $x$ is given by the function $R(x)=\sqrt{\frac{13+7 x^{0.4}}{1+4 x^{0.4}}} $ where $R$ is measured in millimeters and $x$ is measured in appropriate units of brightness.
(a) Find $R(1), R(10),$ and $R(100) .$
(b) Make a table of values of $R(x)$
(c) Find the net change in the radius $R$ as $x$ changes from
10 to 100 .

Key Concepts

Function Evaluation

This concept involves substituting specific numerical inputs into a defined mathematical function and calculating the corresponding outputs. It is a fundamental skill that allows one to understand and predict how changes in the input affect the output. Function evaluation is widely used in various branches of mathematics and applied sciences to analyze behavior and practical implications of models.

Constructing a Table of Values

Creating a table of values is an effective way to organize and visualize the relationship between the input variables and the corresponding outputs of a function. This process assists in identifying patterns or trends, and it is particularly useful for graphing functions or confirming the correctness of computations. It simplifies the analysis of how a function behaves over a range of inputs.

Calculating Net Change in a Function

Net change refers to the overall difference in the value of a function between two given inputs. It is calculated by subtracting the value of the function at the starting input from the value at the ending input. This concept is crucial in many areas such as calculus and applied mathematics, as it helps quantify the cumulative effect of changes in the independent variable over a specified interval.

Transcript

00:01 Given the function r of x, for part a, we're asked to find a few values, a few inputs.

00:11 So first one, r of one would be the square root of 13 plus 7 times 1 to the point 4 over 1 plus 4 times 1 to the 1 to 0 .4, which is equal to the square root of 20 over the square root of, excuse me, the square of 20 over 5.

00:28 Because one of the point four is one this really uses 13 plus 7 which is 20 one of the point four again is one plus four or five 20 over five is four so it's really square of four which is two so this is two millimeters our 10 would be the square root of 13 plus seven times 10 to the point four power over one plus four times 10 to the point four power here we need a calculator once you evaluate what's underneath the square root you get about 2 .7683 which is roughly 1 .66 millimeters and we're asked to find r of 10 excuse me r of 100 which would be found by plugging 100 in for x0 13 plus 7 times 100 to the point of 4 power over 1 plus 4 times 100 to the point 4 power which is roughly the square root of 2 .1787 or about 1 .48 millimeters.

01:35 For part b, a table of values, i'm going to jump over to the decimals online graphy calculator and put this function in as we see it.

01:45 So square root feature, the keypad is in the bottom left.

01:49 So the square root of 13 plus 7x to the 0 .4, and then if you write arrow and then divide, all right, looks like we're going to need parentheses.

02:10 So we want all of this to be in the square root, that's a fraction, perfect.

02:20 Over 1 plus 4x to the point for power.

02:32 All right.

02:35 Once we have the function in, a table of values can be quickly achieved by hitting the cog wheel.

02:40 Let's say convert to table.

02:42 It doesn't say for what value.

02:44 I'm just going to go 0 to 10...