Problem 35 - Calculus

Question

35 \(f(x)\) can be its own derivative. In the infinite polynomial \(f=1+x+\frac{1}{2} x^{2}+\frac{1}{6} x^{3}+\) _______ ,what numbers multiply \(x^{4}\) and \(x^{5}\) if \(d f / d x\) equals \(f\) ?

Step-by-Step Solution

Answer

The coefficients are \( \frac{1}{24} \) for \( x^4 \) and \( \frac{1}{120} \) for \( x^5 \).

Step 1: Identifying the Series Pattern

First, recognize that the function given is a series expansion. The coefficients of this series resemble those of the exponential function's Taylor series expansion, which is: \[ e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!} + \cdots \] Thus, each term of the function is given by \( \frac{x^n}{n!} \).

Step 2: Writing the Derivative

Recall that the derivative of \( e^x \) is itself \( e^x \). Hence, for the series \( f \), the derivative of \( f \) should also follow this pattern. We differentiate the series term by term as: \[ \frac{d}{dx}(1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!} + \cdots) \Equals 0 + 1 + \frac{2x}{2!} + \frac{3x^2}{3!} + \frac{4x^3}{4!} + \frac{5x^4}{5!} + \cdots \]

Step 3: Understanding the Relationship

Since it is given that \( \frac{df}{dx} = f \), we compare the coefficients of the corresponding terms in the original series \( f \) and its derivative:- For \( x^4 \), \( f = \frac{x^4}{4!} \) and from derivative, \( \frac{4x^3}{4!} \).- For \( x^5 \), \( f = \frac{x^5}{5!} \) and from derivative, \( \frac{5x^4}{5!} = \frac{x^4}{4!} \). This suggests the coefficients should maintain this factorial relationship.

Step 4: Solving for Coefficients

The coefficients for \( x^4 \) and \( x^5 \) in \( f \) remain consistent with the pattern established by the Taylor series expansion of \( e^x \):- The coefficient for \( x^4 \) is \( \frac{1}{4!} = \frac{1}{24} \).- The coefficient for \( x^5 \) is \( \frac{1}{5!} = \frac{1}{120} \). Thus, the values that multiply these terms are \( \frac{1}{24} \) and \( \frac{1}{120} \), respectively.

Key Concepts

Exponential Function

The exponential function, denoted as \( e^x \), is a fundamental mathematical function where the constant \( e \) (approximately 2.718) is raised to the power of \( x \). It is unique due to its numerous interesting properties, one of which is that its derivative is the same as the function itself. This means when you differentiate \( e^x \), you still get \( e^x \). Moreover, the exponential function is widely used in diverse fields such as calculus, economics, and natural sciences. Its series expansion is extremely useful for approximations, especially for small values of \( x \). Its series, also known as the Taylor series expansion around 0, is written as: \[ e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots \] This infinite series allows people to understand exponential growth and decay efficiently in various scientific contexts.

Derivatives

In calculus, a derivative represents an instantaneous rate of change of a function concerning one of its variables. For any function \( f(x) \), the derivative, noted as \( f'(x) \) or \( \frac{df}{dx} \), describes how the function's output value changes as its input value shifts. A fascinating property of the exponential function, \( e^x \), is that it is its own derivative. This means:

• The slope or rate of change at any point \( x \) on the curve of \( e^x \) is precisely equivalent to \( e^x \) at that point.
• This characteristic is why \( e^x \) shows up so frequently in growth processes.

To find the derivative of a power series, like the one outlined in our exercise, you take the derivative of each term individually. So: \[ \frac{d}{dx}(1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots) = 0 + 1 + \frac{2x}{2!} + \frac{3x^2}{3!} + \cdots \] calculating the slope of each term as necessary.

Infinite Series

An infinite series is essentially a sum of infinitely many terms. It is represented as the sum of sequence terms, each one being added in an endless progression. In mathematical notations, it's often written as: \[ a_1 + a_2 + a_3 + a_4 + \cdots \] In the context of our function, \( f(x) \) is an example of such an infinite series. It reflects the continuous sum of its terms formed by powers of \( x \) scaled by the reciprocal of factorial numbers. The Taylor series is a specific kind of infinite series used to approximate functions, especially in cases where they can't be easily calculated otherwise. The Taylor series expansion of \( e^x \) provides us an infinite series where each term can be described by:

• \( \frac{x^n}{n!} \), where \( n \) begins from 0 and extends to infinity.
• This representation converges to the function \( e^x \) for all real numbers \( x \).
• Mathematicians and scientists use these expansions for their numerical methods and simulations.

Factorials

Factorials are an essential mathematical concept denoted by the symbol "!". For any integer \( n \), \( n! \) represents the product of all positive integers less than or equal to \( n \). Therefore, \( n! = n \times (n-1) \times (n-2) \times \cdots \times 1 \). For example, \( 4! \) equals \( 4 \times 3 \times 2 \times 1 \), which is 24. Factorials are integral to understanding series expansions, particularly Taylor series and the series for the exponential function:

• In this series, factorials occur in the denominators: \( \frac{x^n}{n!} \).
• They ensure convergence by ensuring each term becomes sufficiently small as \( n \) grows larger.
• This pattern allows \( e^x \) series expressions to consider infinitely many terms while retaining finite results.

Understanding factorials helps in comprehending the distribution and behavior of coefficients within power series and allows calculations to emphasize precision while still being breathable by computational methods.