Problem 25 - Calculus

Question

Find \(a, b, c\) so that \(y=a x^{2}+b x+c\) equals 1,3,7 at \(x=0, \frac{1}{2}, 1\) (three equations). Check that \(\frac{1}{6} \cdot 1+\frac{4}{6} \cdot 3+\frac{1}{6} \cdot 7\) equals \(\int_{0}^{1} y d x\).

Step-by-Step Solution

Answer

Solution involves solving the three linear equations step by step.

Step 1: Set up equations

\(y(0) = c = 1\)
\(y(\frac{1}{2}) = \frac{a}{4} + \frac{b}{2} + c = 3\)
\(y(1) = a + b + c = 7\)

Step 2: Solve the system

From equation 1: \(c = 1\).
Equation 2: \(\frac{a}{4} + \frac{b}{2} + 1 = 3 \Rightarrow \frac{a}{4} + \frac{b}{2} = 2 \Rightarrow a + 2b = 8\).
Equation 3: \(a + b + 1 = 7 \Rightarrow a + b = 6\).
Subtract: \(b = 2\), then \(a = 4\).

Step 3: Verify Simpson's rule

\(y = 4x^2 + 2x + 1\).
\(\frac{1}{6}(1) + \frac{4}{6}(3) + \frac{1}{6}(7) = \frac{1+12+7}{6} = \frac{20}{6} = \frac{10}{3}\).
\(\int_0^1 (4x^2+2x+1)dx = [\frac{4x^3}{3}+x^2+x]_0^1 = \frac{4}{3}+1+1 = \frac{10}{3}\). \(\checkmark\)

Key Concepts

Quadratic Functions

Quadratic functions play a crucial role in mathematics and represent a wide range of real-world phenomena. A quadratic function takes the form of \( y = ax^2 + bx + c \), where \( a, b, \) and \( c \) are constants and \( a eq 0 \). Here, \( a \) affects the curvature of the parabola. If \( a > 0 \), the parabola opens upwards; if \( a < 0 \), it opens downwards. The \( b \) value affects the position of the vertex and the parabola along the x-axis, while \( c \) is the y-intercept, indicating where the parabola crosses the y-axis.

When solving problems involving quadratic functions, a common approach is polynomial fitting, where we seek to determine the coefficients \( a, b, \) and \( c \) so that the function fits given data points. This involves setting up equations based on the given conditions, as seen in this exercise, to enable us to solve for the unknowns.

System of Equations

A system of equations consists of multiple equations that are solved together because they share common variables. In this exercise, each equation represents a condition that the quadratic function must satisfy at specific points. For the equations given:

• At \( x = 0 \), \( c = 1 \)
• At \( x = \frac{1}{2} \), \( \frac{a}{4} + \frac{b}{2} + c = 3 \)
• At \( x = 1 \), \( a + b + c = 7 \)

The goal is to find values of \( a, b, \) and \( c \) that satisfy all these conditions simultaneously.

Solving these equations requires substituting known values into the equations and using algebraic manipulation. This is called 'solving a system of equations' and can be done using methods like substitution, elimination, or matrices. Ultimately, it helps us determine the values that allow the quadratic function to pass through the specified points.

Numerical Integration

Numerical integration involves approximating the integral of a function, especially useful when dealing with functions that are difficult to integrate analytically. It is particularly useful in various fields such as physics, engineering, and statistics where exact answers may be difficult due to complex functions.

In this exercise, numerical integration is used to verify the solution by checking that the polynomial's integral over a given interval \([0, 1]\) matches a weighted sum of the function's values at specified points. This is akin to using a method like the trapezoidal rule, where we approximate the area under the curve. The weighted sum \( \frac{1}{6} \times 1 + \frac{4}{6} \times 3 + \frac{1}{6} \times 7 \) is expected to be equal to the integral \( \int_{0}^{1} y \: dx \).

By understanding numerical integration, students gain insights into approximation techniques and their practical applications in accurately estimating integrals for real-world data, particularly when an exact integral is challenging to evaluate.