/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 24 If \(y(n)=e^{x} e^{x^{2}} \ldots... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 24

If \(y(n)=e^{x} e^{x^{2}} \ldots e^{x^{n}}, 0

Short Answer

Expert verified
The short answer is: (c) \(2 e\).

Step by step solution

01

Write the given function in a compact form

We can write the given function y(n) as: \[y(n) = e^{x} e^{x^{2}} \ldots e^{x^{n}}\] Simplify the expression using the exponent addition rule: \[y(n) = e^{(x + x^2 + \dots + x^n)}\]
02

Calculate the derivative of y(n) with respect to x

Now, let's differentiate y(n) with respect to x using the chain rule: \[\frac{dy(n)}{dx} = e^{(x + x^2 + \dots + x^n)} \cdot \frac{d(x + x^2 + \dots + x^n)}{dx}\] Next, differentiate the sum of x terms: \[\frac{d(x + x^2 + \dots + x^n)}{dx} = 1 + 2x + \dots + nx^{n-1}\] So, the derivative of y(n) w.r.t x becomes: \[\frac{dy(n)}{dx} = e^{(x + x^2 + \dots + x^n)} (1 + 2x + \dots + nx^{n-1})\]
03

Calculate the limit

Now, we need to find the limit of the derivative as n approaches infinity: \[\lim_{n \to \infty} \frac{dy(n)}{dx}\] Notice that the factor (1 + 2x + ... + nx^{n-1}) is finite and does not depend on n. Thus, we focus on the limit of the exponent part.
04

Evaluate the limit

We must evaluate the following limit: \[\lim_{n \to \infty} e^{(x + x^2 + \dots + x^n)}\] Since 0 < x < 1, the sum of the exponents converges to a finite value. Therefore, the limit becomes: \[\lim_{n \to \infty} e^{(x + x^2 + \dots + x^n)} = e^S\] where S is the sum of the convergent series (x + x^2 + ... ). We can also proceed to compute S and evaluate the derivative at x = 1/2.
05

Evaluate at x = 1/2

Plug in x = 1/2 in the derivative: \[\frac{dy(n)}{dx} = e^{(0.5 + 0.25 + \dots + (0.5)^n)} (1 + 2(0.5) + \dots + n(0.5)^{n-1})\] Now, find the limit of the function as n approaches infinity: \[\lim_{n \to \infty} \frac{dy(n)}{dx} = e^{\sum_{k=1}^{\infty} \frac{1}{2^k}} \left(\sum_{k=1}^{\infty} k \frac{1}{2^{k}}\right)\] Using geometric series properties, and noting that both series converge, we obtain: \[ e^{\frac{1/2}{1-1/2}} \cdot \frac{1/4}{(1/2)^2} = e^1\cdot2 = 2e\] Hence, the correct answer is (c) \(2 e\).

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Key Concepts

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

Limit of a Function
In calculus, the concept of limits helps us understand the behavior of functions as they approach a specific point or as a variable approaches infinity. The goal is to find out what the function is heading towards, even if it never actually reaches that point.
  • For a function like \( f(x) \), the limit as \( x \) approaches a point \( a \) is denoted as \( \lim_{{x \to a}} f(x) \).
  • In some cases, like in this exercise, we are also interested in the limit as \( n \to \infty \), which means we want to know how the function behaves when \( n \) gets very large.
For the given exercise, we analyzed the `limit` of the derivative of the function as \( n \) approaches infinity. By understanding both the convergence of the exponential series and the characteristic properties of geometric series, we were able to compute a defined finite limit.
Derivative
The derivative of a function at a point is a measure of the rate at which the function's value changes as its input changes. It is like the slope of a line at any given point on the function's graph.
  • The derivative of a function \( y = f(x) \) concerning \( x \) is usually represented as \( \frac{dy}{dx} \).
  • To find a derivative, we often use rules like the power rule, product rule, or chain rule depending on the complexity of the function.
In this particular exercise, we used the chain rule to differentiate the exponential function \( y(n) = e^{(x + x^2 + \ldots + x^n)} \). Knowing how to differentiate allowed us to express how \( y(n) \) changes with \( x \), which was crucial for finding the limit.
Geometric Series
A geometric series is a series with a constant ratio between successive terms. Understanding geometric series is valuable as it greatly simplifies the calculations of certain sums.
  • A geometric series can be written generally as \( a + ar + ar^2 + \ldots \), where \( r \) is the common ratio.
  • If the absolute value of \( r \) is less than 1, the series converges to \( \frac{a}{1-r} \).
In the exercise, we handled a series with terms like \( x, x^2, \ldots, x^n \) by identifying it as a geometric series where \( x \) is the common ratio. This helps find the sum of the series as \( n \to \infty \), using the formula for convergent geometric series.
Exponent Rules
Exponent rules help simplify expressions with exponents and are essential in solving derivatives and limits involving exponential functions. Here are some crucial exponent rules:
  • The product rule: \( a^m \times a^n = a^{m+n} \).
  • The power of a product rule: \( (ab)^n = a^n \times b^n \).
  • The power of a power rule: \( (a^m)^n = a^{m \times n} \).
In this exercise, we had multiple exponential terms like \( e^x \), \( e^{x^2} \), ..., \( e^{x^n} \). Using the product rule, we condensed these into a single exponential expression \( e^{(x + x^2 + \ldots + x^n)} \). This simplification was key to effectively differentiating and evaluating the limit of the function.

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Most popular questions from this chapter

If \(\quad y=\frac{1}{\sqrt{a^{2}-b^{2}-c^{2}}} \cos ^{-1}\left\\{\frac{a \theta-a^{2}+b^{2}+c^{2}}{\theta \sqrt{b^{2}+c^{2}}}\right\\}\) and \(\theta=a+b \cos x+c \sin x ;\) prove that \(\frac{d y}{d x}=\frac{1}{\theta}\)

Let \(f\) and \(g\) be differentiable function such that \(f^{\prime}(x)=2 g(x)\) and \(g^{\prime}(x)=-f(x)\), and let \(T(x)=\) \((f(x))^{2}-(g(x))^{2}\). Then \(T^{\prime}(x)\) is equal to (a) \(T(x)\) (b) 0 (c) \(2 f(x) g(x)\) (d) \(6 f(x) g(x)\)

If \(f(x+y)=f(x)+f(y) \forall, x, y\) and \(f(0)\) exists., then (a) \(f^{\prime}(x)=f^{\prime}(0) \forall x\) (b) \(f(x)=c x\), where \(c\) is a constant (c) \(f(x)=x f(1)\) (d) \(f(x)=\frac{x}{2} f(2)\)

If \(\int_{\pi / 2}^{x} \sqrt{\left(5+2 \sin ^{2} t\right)}+\int_{0}^{y} \cos t d t=0\), then \(\left(\frac{d y}{d x}\right)_{\pi, \pi}\) equal to (a) \(-5\) (b) 0 (c) \(\sqrt{5}\) (d) None of these

Given a function g which has derivative \(\quad h^{\prime}(x) \forall x\) satisfying \(h^{\prime}(0)=2 \quad\) and \(h(x+y)=e^{y} h(x)+e^{x} h(y) \forall x, y \in \mathbb{R}, h(5)=32\). The value of \(h^{\prime}(5)-2 e^{5}\) is
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