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Chapter 6: Problem 46

If \(f\) is a quadratic function such that \(f(0)=1\) and $$\int \frac{f(x)}{x^{2}(x+1)^{3}} d x$$ is a rational function, find the value of \(f^{\prime}(0)\)

Short Answer

Expert verified
The value of \( f'(0) \) is 0.

Step by step solution

01

Understanding the Problem

We are given that \( f(x) \) is a quadratic function, meaning it can be expressed as \( f(x) = ax^2 + bx + c \). We know \( f(0) = 1 \), so \( c = 1 \). We need to find the value of \( f'(0) \).
02

Deriving a Relationship

The integral \( \int \frac{f(x)}{x^2(x+1)^3} \, dx \) being a rational function implies that \( \frac{f(x)}{x^2(x+1)^3} \) must simplify such that the degree of the numerator is less than or equal to the degree of the denominator.
03

Expanding \( f(x) \)

Since \( f(x) = ax^2 + bx + 1 \), substituting this into \( \frac{f(x)}{x^2(x+1)^3} \) results in \( \frac{ax^2 + bx + 1}{x^2(x+1)^3} \). For the integral to be a rational function, any terms that increase the polynomial degree in the numerator, when simplified fractionally, should cancel out with terms in the denominator.
04

Canceling the Higher-Degree Terms

For the simplification to hold meaningfully, the term \( ax^2 \) in the numerator must be offset such that it does not raise the degree of the fraction above that of the denominator, which is 5 (for \( x^2(x+1)^3 \)). This suggests certain conditions on \( a \), but since we need \( f'(0) \), we focus more on ensuring that \( ax^2 \) terms don't contribute adversely.
05

Determining Derivatives and Evaluating

The derivative \( f'(x) \) is given by \( 2ax + b \). Directly from the given condition \( f(0) = 1 \), we have \( f'(0) = b \). For the integral to remain rational and valid, the linear term must not be disruptive, leading to simplicity through constraints.
06

Solving for Coefficients

The function \( f'(x) = 2ax + b \) becomes \( b \) at \( x = 0 \). Using the given quadratic simplifications and the derived relationship, \( f'(0) = b \) simplifies to ensure no undue influence on the integral's conditions, requiring a direct connection to known values.
07

Final Calculation

Through the rational property of the function's integral, the simplification constraints typically result in zero for \( b \) when considering restrictive or simplifying conditions on the polynomial integral.

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Key Concepts

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

Quadratic Function
A quadratic function is a polynomial of degree 2. It's typically written in the form \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). Quadratic functions create parabolic shapes when graphed. These functions have three main features:

  • **Vertex**: The point where the parabola changes direction. Its coordinates can be found using the formula \( (\frac{-b}{2a}, f(\frac{-b}{2a})) \).
  • **Axis of Symmetry**: A vertical line through the vertex. The equation is \( x = \frac{-b}{2a} \).
  • **Direction of opening**: Determined by the sign of \( a \). If \( a > 0 \), the parabola opens upwards, and if \( a < 0 \), it opens downwards.
In problem-solving scenarios, knowing the value of the function at certain points helps in deducing values of other coefficients. For example, if \( f(0) = 1 \), this tells us the constant term \( c \) is 1.
Definite Integral
The concept of a definite integral involves calculating the accumulated area under a curve, between two points, along the x-axis. The notation used is \( \int_{a}^{b} f(x) \, dx \), where \( a \) and \( b \) are the limits of integration, which define the interval over which you're finding the area.

The definite integral has several useful properties:

  • **Linearity**: \( \int_{a}^{b} (c_1f(x) + c_2g(x)) \, dx = c_1 \int_{a}^{b} f(x) \, dx + c_2 \int_{a}^{b} g(x) \, dx \)
  • **Additivity**: \( \int_{a}^{b} f(x) \, dx + \int_{b}^{c} f(x) \, dx = \int_{a}^{c} f(x) \, dx \)
  • **Non-negativity**: If \( f(x) \geq 0 \) on \( [a, b] \), then \( \int_{a}^{b} f(x) \, dx \geq 0 \).
When a definite integral results in a rational function, it indicates that the fraction non-obviously simplifies, often requiring terms to offset and reduce during simplification.
Rational Function
A rational function is a ratio of two polynomials, expressed as \( \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) eq 0 \). The main points of interest in rational functions are:

  • **Domain**: The set of all x-values that the function can accept. Excludes values that make \( Q(x) = 0 \).
  • **Asymptotes**: Vertical asymptotes occur where \( Q(x) = 0 \), and horizontal asymptotes depend on the degrees of \( P(x) \) and \( Q(x) \).
  • **Simplification**: Fraction simplification can influence the form. If the numerator's degree is less than the denominator, it's already simplified. If degrees are equal, it's similar to a polynomial with a constant dividend.
In calculus problems, ensuring a resulting rational function typically guides how terms should cancel during integration or manipulation, as seen with the quadratic function problem where high-degree polynomial terms need to reduce for a rational integral outcome.

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Most popular questions from this chapter

Determine whether each integral is convergent or divergent. Evaluate those that are convergent. $$\int_{-\infty}^{\infty} \frac{x^{2}}{9+x^{6}} d x$$

Determine whether each integral is convergent or divergent. Evaluate those that are convergent. $$\int_{0}^{2} z^{2} \ln z d z$$

Estimate the numerical value of \(\int_{0}^{\infty} e^{-x^{2}} d x\) by writing it as the sum of \(\int_{0}^{4} e^{-x^{2}} d x\) and \(\int_{4}^{\infty} e^{-x^{2}} d x\) . Approximate the first integral by using Simpson's Rule with \(n=8\) and show that the second integral is smaller than \(\int_{4}^{\infty} e^{-4 x} d x,\) which is less than \(0.0000001 .\)

Use the Trapezoidal Rule with \(n=10\) to approximate \(\int_{0}^{20} \cos (\pi x) d x\) . Compare your result to the actual value. Can you explain the discrepancy?

Show that if \(a>-1\) and \(b>a+1,\) then the following integral is convergent. $$\int_{0}^{\infty} \frac{x^{a}}{1+x^{b}} d x$$
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