Problem 7 - Applied Calculus

Question

The size, \(S\), of a tumor (in cubic millimeters) is given by \(S=2^{t}\), where \(t\) is the number of months since the tumor was discovered. Give units with your answers. (a) What is the total change in the size of the tumor during the first six months? (b) What is the average rate of change in the size of the tumor during the first six months? (c) Estimate the rate at which the tumor is growing at \(t=6\). (Use smaller and smaller intervals.)

Step-by-Step Solution

Answer

(a) 63 mm³; (b) 10.5 mm³/month; (c) 64 mm³/month.

Step 1: Calculate Tumor Size at Month 0

The size of the tumor at month 0, when the tumor was first discovered, is given by substituting \(t=0\) into the size formula: \[ S = 2^{0} = 1 \text{ mm}^3 \]

Step 2: Calculate Tumor Size at Month 6

To find the size of the tumor at the end of six months, substitute \(t=6\) into the formula:\[ S = 2^{6} = 64 \text{ mm}^3 \]

Step 3: Compute Total Change in Tumor Size

The total change in the size of the tumor from month 0 to month 6 is given by subtracting the size at \(t=0\) from the size at \(t=6\):\[ \Delta S = 64 - 1 = 63 \text{ mm}^3 \]

Step 4: Calculate Average Rate of Change Over Six Months

The average rate of change in the tumor size over the first six months is the total change in size divided by the change in time:\[ \text{Average Rate} = \frac{63 \text{ mm}^3}{6 \text{ months}} = 10.5 \text{ mm}^3/\text{month} \]

Step 5: Estimate Instantaneous Growth Rate at Month 6

To estimate the growth rate at \( t=6 \), calculate the change in size over increasingly smaller time intervals around \( t=6 \), such as \( [5.9, 6] \), \( [5.99, 6] \), and \( [5.999, 6] \).1. For \( \Delta t = 0.1 \): \( S(5.9) = 2^{5.9} \approx 60.715 \text{ mm}^3 \) \( S(6) = 64 \text{ mm}^3 \) \( \text{Rate} = \frac{64 - 60.715}{0.1} = 32.85 \text{ mm}^3/\text{month} \)2. For \( \Delta t = 0.01 \): \( S(5.99) = 2^{5.99} \approx 63.740 \text{ mm}^3 \) \( \text{Rate} = \frac{64 - 63.740}{0.01} = 26 \text{ mm}^3/\text{month} \)3. For \( \Delta t = 0.001 \): \( S(5.999) = 2^{5.999} \approx 63.936 \text{ mm}^3 \) \( \text{Rate} = \frac{64 - 63.936}{0.001} = 64 \text{ mm}^3/\text{month} \)As \( \Delta t \to 0 \), the rate approaches \( 64 \text{ mm}^3/\text{month} \).

Key Concepts

Tumor Size

Tumor size refers to the volume that a tumor occupies within the body, and it can be an essential factor in determining the stage and seriousness of a disease like cancer. In our exercise, the tumor size, denoted as \( S \), is given by the equation \( S = 2^t \). This equation illustrates how the size changes over time, measured in months \( t \) from the time of discovery.

• Initially, at \( t = 0 \), the size is \( S = 2^0 = 1 \text{ mm}^3 \). This represents the tumor's size when first discovered.
• After six months, when \( t = 6 \), the size grows to \( S = 2^6 = 64 \text{ mm}^3 \). This significant increase exemplifies exponential growth, a key concept in analyzing tumor progression.

Understanding the tumor size's exponential growth helps in anticipating future changes and planning treatment strategies effectively.

Rate of Change

The rate of change is a mathematical term used to describe how a quantity changes over a certain period. In this context, it relates to how quickly the tumor size changes over time. This is crucial for healthcare providers, as understanding the rate of change can dictate treatment urgency and type.

• A positive rate of change indicates that the tumor is increasing in size, which is the case in our exercise due to its exponential growth.

Understanding how quickly a tumor grows is essential for detecting aggressive cases early and adapting treatment plans accordingly.

Average Rate of Change

The average rate of change of the tumor size over a specified period provides a summary of the overall growth speed. It is calculated by dividing the total change in size by the time elapsed.

• For the first six months, the total size change is \( 63 \text{ mm}^3 \) (growth from \( 1 \text{ mm}^3 \) to \( 64 \text{ mm}^3 \)).
• To find the average, divide this total growth by the 6 months: \( \frac{63 \text{ mm}^3}{6 \text{ months}} = 10.5 \text{ mm}^3/\text{month} \).

This average provides a simplified view of the tumor's growth rate over the entire period, which is helpful for tracking and comparing growth in different time ranges.

Instantaneous Rate of Change

The instantaneous rate of change refers to the rate at which the tumor size is increasing at an exact moment, such as at the 6-month mark in our scenario. To estimate this rate, we assess how the size changes over increasingly smaller time intervals approaching the specific point of interest like \( t = 6 \).

• We start by calculating the size just before 6 months, like at \( t = 5.999 \), resulting in a more accurate estimate of the change close to \( t = 6 \).
• For \( \Delta t = 0.001 \), the rate is computed as \( 64 - 63.936 = 0.064 \text{ mm}^3 \), which gives \( 64 \text{ mm}^3/\text{month} \).

By letting the time interval tend to zero, we discover the instantaneous rate at exactly 6 months, representing the exact growth rate at that moment. This precision is vital in fine-tuning medical treatment responses.