Problem 29 - Applied Calculus

Question

For some painkillers, the size of the dose, \(D\), given depends on the weight of the patient, \(W\). Thus, \(D=f(W)\), where \(D\) is in milligrams and \(W\) is in pounds. (a) Interpret the statements \(f(140)=120\) and \(f^{\prime}(140)=3\) in terms of this painkiller. (b) Use the information in the statements in part (a) to estimate \(f(145)\).

Step-by-Step Solution

Answer

The dose for 140 pounds is 120 mg, with a 3 mg increase per pound; estimate 135 mg for 145 pounds.

Step 1: Understanding f(140)=120

The function value \(f(140) = 120\) means that for a patient weighing 140 pounds, the appropriate dosage of the painkiller is 120 milligrams. In other words, when the patient's weight is 140 pounds, they should receive a 120 mg dose.

Step 2: Understanding f'(140)=3

The derivative \(f'(140) = 3\) signifies that at a weight of 140 pounds, the dosage increases by 3 milligrams for each additional pound of body weight. This tells us the rate of change of the dosage with respect to weight at this point.

Step 3: Estimating f(145) Using Linear Approximation

To estimate \(f(145)\), we use the linear approximation: \(f(a+h) \approx f(a) + h \cdot f'(a)\). Here, \(a = 140\) and \(h = 5\).\[ f(145) \approx f(140) + 5 \cdot f'(140) = 120 + 5 \times 3 = 135. \] So, the estimated dosage for a patient weighing 145 pounds is 135 milligrams.

Key Concepts

Function Interpretation

When we talk about function interpretation in mathematics, we're essentially trying to understand what a particular function tells us about the real world. In our exercise, we have a function, \( D = f(W) \), which relates a patient's weight, \( W \), to their required dosage of a painkiller, \( D \).

The statement \( f(140) = 120 \) means that when a patient weighs 140 pounds, they need a dose of 120 milligrams of the painkiller. This is a specific point on the graph of the function, showing the relationship between weight and dosage at this weight.

Interpreting these numbers can be very practical, especially in healthcare, where accurate dosages are crucial to safety and effectiveness. It tells us precisely what dosage to administer based on the given weight.

Linear Approximation

Linear approximation is a method used in calculus to approximate the value of a function near a given point using its derivative. This is also known as a tangent line approximation because it uses the slope of the tangent line to estimate changes in the function.

We use a formula for linear approximation: \[f(a+h) \approx f(a) + h \cdot f'(a)\] In our problem, \( a = 140 \), \( h = 5 \), \( f(a) = 120 \), and \( f'(a)=3 \). This helps us estimate \( f(145) \) quickly:

• Start with \( f(140) = 120 \).
• Calculate \( h \times f'(140) = 5 \times 3 = 15 \).
• Add this to the original function value: \( 120 + 15 = 135 \).

Thus, the estimated painkiller dose for a 145-pound patient is 135 mg. This technique simplifies calculations and is particularly useful when exact values are difficult to find.

Rate of Change

The rate of change tells us how one quantity changes in response to changes in another quantity. In calculus, this is often referred to as the derivative.

In our problem, the derivative \( f'(140) = 3 \) indicates that for every additional pound of body weight above 140 pounds, the dosage should increase by 3 milligrams. This is a key insight because it allows for adjustments to dosage as weight changes.

Understanding the rate of change is crucial because:

• It guides us in making predictions about future values.
• It helps in understanding how sensitive a system or model is to changes.
• In healthcare, it ensures that each patient receives the correct medication dosage based on specific parameters like weight.

Using the rate of change makes it possible for health professionals to provide personalized treatments.