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Chapter 1: Problem 59

A drug is administered to a patient through an IV (intravenous) injection at the rate of 6 milliliters (mL) per minute. Assuming that the patient's body already contained \(1.5 \mathrm{mL}\) of this drug at the beginning of the infusion, find an expression for the amount of the drug in the body \(x\) minutes from the start of the infusion.

Short Answer

Expert verified
The amount of the drug in the body x minutes from the start of the infusion is given by the expression \( 1.5 + 6x \).

Step by step solution

01

Identify the Initial Amount

Determine the initial amount of the drug already present in the patient's body. This amount is given as 1.5 milliliters (mL).
02

Determine the Rate of Infusion

Identify the rate at which the drug is being administered. In this case, the rate is given as 6 milliliters per minute (mL/min).
03

Establish a Time Variable

Let the variable x represent the number of minutes from the start of the infusion.
04

Create the Expression for the Additional Drug

The amount of drug added after x minutes can be calculated using the rate of infusion. This is given by the expression: \[ 6x \]
05

Combine Initial Amount with Infusion Amount

Add the initial amount of the drug to the amount added through the infusion to find the total amount of the drug in the body. Thus, the expression becomes: \[ 1.5 + 6x \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rate of Change
The 'rate of change' is a fundamental concept in calculus and other mathematical disciplines. It refers to how a quantity changes over time. In this exercise, the rate of change is \(6 \text{ mL/min}\), representing how quickly the drug is being administered into the patient's body.

We often express the rate of change with variables such as \(6x\), where \(x\) is the time in minutes. This helps us calculate how much of the drug is added with each passing minute. In general:
- A constant rate means a linear change.
- A variable rate means a non-linear change.

Understanding the rate of change helps us predict future amounts, like how much drug will be in the patient after a certain number of minutes.
Initial Value Problem
An 'initial value problem' (IVP) is a specific type of problem in calculus where you know the initial state or value of a quantity and need to find how it changes over time. In our drug administration exercise, the initial value is \(1.5 \text{ mL}\) of the drug already present in the patient's body at the beginning.

To solve an IVP, you:
- Identify the initial amount (given as \(1.5 \text{ mL}\) here).
- Establish a formula for how the amount changes over time (resolved by combining initial value and rate of change).

For example, if \(1.5 \text{ mL}\) is already in the body and \(6 \text{ mL}\) is being added every minute, we find the total amount of drug using the formula: $$ 1.5 + 6x $$

The initial value sets the starting point for our calculations and predictions.
Linear Functions
A 'linear function' is one in which the relationship between two variables can be represented by a straight line. Its general form is: $$ y = mx + c $$
Here, \(m\) is the slope representing the rate of change, while \(c\) is the y-intercept signifying the initial value.

In our exercise, the amount of drug after \(x\) minutes is a linear function: $$ 1.5 + 6x $$

- \(6\) is the rate at which the drug is infused (slope).
- \(1.5\) is the initial amount of the drug (intercept).

Linear functions are a powerful way to model real-world situations where changes happen at a constant rate. They are simple to work with and make it easy to understand and predict trends over time.

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Most popular questions from this chapter

Determine which of the following limits exist. Compute the limits that exist. Compute the limits that exist, given that $$\lim _{x \rightarrow 0} f(x)=-\frac{1}{2} \quad\( and \)\quad \lim _{x \rightarrow 0} g(x)=\frac{1}{2}.$$ (a) \(\lim _{x \rightarrow 0}(f(x)+g(x))\) (b) \(\lim _{x \rightarrow 0}(f(x)-2 g(x))\) (c) \(\lim _{x \rightarrow 0} f(x) \cdot g(x)\) (d) \(\lim _{x \rightarrow 0} \frac{f(x)}{g(x)}\)

Let \(S(x)\) represent the total sales (in thousands of dollars) for the month \(x\) in the year 2005 at a certain department store. Represent each following statement by an equation involving \(S\) or \(S^{\prime}\) (a) The sales at the end of January reached 120,560 dollars and were rising at the rate of 1500 dollars per month. (b) At the end of March, the sales for this month dropped to 80,000 dollars and were falling by about 200 dollars a day. (Use 1 month \(=30\) days.

A toy company introduces a new video game on the market. Let \(S(x)\) denote the number of videos sold on the day, \(x,\) since the item was introduced. Let \(n\) be a positive integer. Interpret \(S(n), S^{\prime}(n),\) and \(S(n)+S^{\prime}(n)\)

Find the slope of the tangent line to the curve \(y=x^{3}+3 x-8\) at \((2,6).\)

Let \(y\) denote the percentage of the world population that is urban \(x\) years after 2014. According to data from the United Nations, 54 percent of the world's population was urban in 2014 , and projections show that this percentage will increase to 66 percent by 2050. Assume that \(y\) is a linear function of \(x\) since 2014. (a) Determine \(y\) as a function of \(x.\) (b) Interpret the slope as a rate of change. (c) Find the percentage of the world's population that is urban in 2020. (d) Determine the year in which \(72 \%\) of the world's population will be urban.
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