Problem 102 - Thomas Calculus

Question

a. Suppose that \(f(x)\) is differentiable for all \(x\) in \([0,1]\) and that \(f(0)=0 .\) Define sequence \(\left\\{a_{n}\right\\}\) by the rule \(a_{n}=n f(1 / n)\) . Show that \(\lim _{n \rightarrow \infty} a_{n}=f^{\prime}(0) .\) Use the result in part (a) to find the limits of the following sequences \(\left\\{a_{n}\right\\} .\) $$\begin{aligned} \text { b. } a_{n} &=n \tan ^{-1} \frac{1}{n} & \text { c. } a_{n}=n\left(e^{1 / n}-1\right) \\ \text { d. } a_{n} &=n \ln \left(1+\frac{2}{n}\right) \end{aligned}$$

Step-by-Step Solution

Answer

The limits are: (b) 1, (c) 1, (d) 2.

Step 1: Understand the Problem

To solve this problem, we need to show that the sequence defined as \(a_n = n f(1/n)\) converges to \(f'(0)\) as \(n\) approaches infinity when \(f\) is differentiable and \(f(0) = 0\). This will then allow us to determine the limits of specified sequences by replacing \(f\) with the respective function.

Step 2: Apply the Definition of Derivative

We begin by recalling the definition of the derivative at a point. The derivative \(f'(0)\) is given by \[ f'(0) = \lim_{h \to 0} \frac{f(h) - f(0)}{h} = \lim_{h \to 0} \frac{f(h)}{h} \] since \(f(0)=0\). Substitute \(h = \frac{1}{n}\) to transform the limit: as \(h \to 0\), \(n \to \infty\). Thus, \[ f'(0) = \lim_{n \to \infty} n f\left(\frac{1}{n}\right) = \lim_{n \to \infty} a_n. \] Hence, \(a_n\) converges to \(f'(0)\).

Step 3: Calculate Limit for Sequence b

For \(a_n = n \tan^{-1}\left(\frac{1}{n}\right)\), consider \(f(x) = \tan^{-1}(x)\). Then \(f'(x) = \frac{1}{x^2 + 1}\), so \(f'(0) = 1\). Therefore, \(\lim_{n \to \infty} a_n = 1\).

Step 4: Calculate Limit for Sequence c

For \(a_n = n(e^{1/n} - 1)\), consider \(f(x) = e^x - 1\). Then \(f'(x) = e^x\), so \(f'(0) = e^0 = 1\). Therefore, \(\lim_{n \to \infty} a_n = 1\).

Step 5: Calculate Limit for Sequence d

For \(a_n = n \ln\left(1+\frac{2}{n}\right)\), consider \(f(x) = \ln(1+2x)\). Then \(f'(x) = \frac{2}{1+2x}\), so \(f'(0) = 2\). Therefore, \(\lim_{n \to \infty} a_n = 2\).

Key Concepts

Sequence Limits

In mathematics, a sequence limit is a fundamental concept in understanding the behavior of sequences as they approach infinity. A sequence \(a_n\) has a limit \(L\) if, for any small positive number \(\epsilon\), there exists a point in the sequence beyond which all terms are within \(\epsilon\) of \(L\). In formal terms: \[\lim_{n \to \infty} a_n = L \]This means that as \(n\) increases indefinitely, \(a_n\) becomes arbitrarily close to \(L\). Sequence limits are crucial in calculus when analyzing the behavior of functions as parameters change over time.

• They help determine stability and increase accuracy in mathematical predictions.
• Sequence limits aid in understanding real-world phenomena described by physical, social, or economic models via calculus.

Having a strong grasp of how to find and understand the limits of sequences enables students to comprehend more complex mathematical concepts efficiently.

Derivative Definition

The derivative of a function is a fundamental concept in differential calculus. It measures how a function changes as its input changes. When we say that a function \(f(x)\) is differentiable at a point \(x = a\), we mean that it has a well-defined derivative at that point. The derivative at a point \(a\) is defined by the limit:\[f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}\]Here, \(h\) represents a small increment in \(x\), allowing us to find how \(f(x)\) behaves as \(x\) changes near \(a\).

• In simpler terms, the derivative tells us the rate of change or the slope of the tangent to the curve at that point.
• The process of finding a derivative is called differentiation.

Differentiation is used across various fields for optimizing and understanding trends, improving machine learning algorithms, and even in medical modeling. It helps predict outcomes and make decisions based on changing variables.

Convergence of Sequences

Convergence of sequences is a critical idea in calculus that discusses whether a sequence approaches a specific value as \(n\) becomes very large. When a sequence \(a_n\) converges, it means that there is some finite limit \(L\) that the terms of the sequence get closer to, as \(n\) increases. Mathematically, a sequence \(a_n\) converges to \(L\) if, for every \(\epsilon > 0\), there is a positive integer \(N\) such that:\[|a_n - L| < \epsilon\quad \text{for all } n > N\]To apply this concept, consider examining sequences for convergence properties:

• It helps in determining steady-state solutions in dynamic systems.
• Convergence is fundamental in computing integrals and solving equations in implicit forms.

Thus, understanding sequence convergence paves the way for tackling more advanced calculus topics and is deeply rooted in mathematical analysis techniques utilized in a wide range of scientific inquiries.