Book cover for Biocalculus Calculus for the Life Sciences

Biocalculus Calculus for the Life Sciences

James Stewart

ISBN #9781133109631

1st Edition

2,565 Questions

Group icon
211,110 Students Helped

Homework Questions

Right arrow
Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section explores the concept of limits for sequences and functions by investigating their long-term behavior. Key topics include the definition of limits, convergence versus divergence, and the application of limit laws to both rational and geometric sequences. Additionally, the section provides practical examples such as drug dosage modeling and the Monod growth function, highlighting the relevance of these ideas in real-world applications such as biology and medicine.

Learning Objectives

1

Explain the concept of the limit of a sequence and distinguish between convergent and divergent sequences.

2

Apply limit laws to determine the long-term behavior of sequences, including geometric sequences and series.

3

Analyze recursive formulas and apply them to real-world problems such as drug concentration and population growth.

4

Understand the modeling of growth functions through the Monod (Michaelis-Menten) function and its applications in biology.

Key Concepts

CONCEPT

DEFINITION

Sequence

An ordered list of numbers defined either by an explicit formula or recursively by a difference equation.

Limit of a Sequence

A value L such that the terms of a sequence approach L as n becomes arbitrarily large. Notation: limₙ→∞ aₙ = L.

Convergent Sequence

A sequence that has a limit; its terms get arbitrarily close to a specific number as n increases.

Divergent Sequence

A sequence that does not approach any fixed number as n increases, or diverges to infinity or oscillates.

Geometric Sequence

A sequence of the form a, ar, ar², ar³,... where a is the initial term and r is the constant ratio between consecutive terms.

Infinite Series

The sum of all terms of an infinite sequence, which may converge (have a finite sum) if the common ratio satisfies |r| < 1.

Logistic Equation

A recursive relation of the form xₜ₊₁ = xₜ + c·xₜ(1 - xₜ) used to model population dynamics and phenomena that may exhibit chaotic behavior.

Monod Growth Function

A function R(N) = (cN)/(S + N) used to model enzyme reactions and bacterial growth, where c is a constant and S represents saturation.

Example Problems

Example 1

(a) What is a sequence? (b) What does it mean to say that $\lim _{n \rightarrow \infty} a_{n}=8 ?$ (c) What does it mean to say that $\lim _{n \rightarrow \infty} a_{n}=\infty ?$

Example 2

(a) What is a convergent sequence? Give two examples. (b) What is a divergent sequence? Give two examples.

Example 3

World record sprint times The graph plots the sequence of the world record times for the men's 100 -meter sprint every five years $t$ . Do you think that this sequence has a nonzero limit as $t \rightarrow \infty ?$ What would that mean for this sporting event?

Example 4

World record hammer throws The graph plots the sequence of the world record distances for the women's hammer throw by year $t$ . (a) Explain what it would mean for this sporting event if the sequence does not have a limit as $t \rightarrow \infty$ . (b) Do you think this sequence is convergent or divergent? Explain.

Example 5

$5-8$ Calculate, to four decimal places, the first ten terms of the sequence and use them to plot the graph of the sequence. Does the sequence appear to have a limit? If so, calculate it. If not, explain why. $a_{n}=\frac{n^{2}}{2 n+3 n^{2}}$
Scroll left
Scroll right

Step-by-Step Explanations

QUESTION

Find limₙ→∞ (1 + 2n²) / (5 + 3n + 4n²).

STEP-BY-STEP ANSWER:

Step 1: Identify the highest power of n in the denominator; here, it is n².
Step 2: Divide both numerator and denominator by n² to get: (1/n² + 2) / (5/n² + 3/n + 4).
Step 3: As n becomes very large, terms with n in the denominator (1/n², 5/n², and 3/n) approach 0.
Step 4: The expression simplifies to 2/4.
Step 5: Compute 2/4 to obtain 0.5.
Final Answer: limₙ→∞ (1 + 2n²) / (5 + 3n + 4n²) = 0.5.

Evaluating the Limit of a Rational Sequence

QUESTION

For a geometric sequence defined by aₙ = a·rⁿ, under what condition does limₙ→∞ aₙ exist and what is it?

STEP-BY-STEP ANSWER:

Step 1: Recognize that the behavior depends on the absolute value of r.
Step 2: If |r| < 1, then as n increases, rⁿ approaches 0.
Step 3: Therefore, limₙ→∞ aₙ = a · 0 = 0.
Step 4: If r = 1, then the sequence is constant and limₙ→∞ aₙ = a.
Step 5: If |r| > 1, the sequence diverges (grows indefinitely or oscillates if r is negative).
Final Answer: The sequence converges to 0 if |r| < 1; converges to a if r = 1; diverges if |r| > 1.

Determining Convergence of a Geometric Sequence

Scroll left
Scroll right

Common Mistakes

  • Forgetting to divide the numerator and denominator by the highest power of n when evaluating limits of rational sequences.
  • Assuming that a sequence always converges without checking the conditions on the common ratio in geometric sequences.
  • Mixing up convergent behavior with general divergence; not recognizing when a sequence does not approach a fixed number.
  • Overlooking the necessity to verify that a limiting value exists before applying limit laws in recursive models.