Problem 90 - Calculus Volume 1

Question

In precalculus, you learned a formula for the position of the maximum or minimum of a quadratic equation \(y=a x^{2}+b x+c, \quad\) which was \(m=-\frac{b}{(2 a)},\) Prove this formula using calculus.

Step-by-Step Solution

Answer

Using calculus, setting the derivative of the quadratic equation to zero gives the extremum at \( x = -\frac{b}{2a} \).

Step 1: Identify the function and its derivative

We start with the quadratic function \( y = ax^2 + bx + c \). To find the maximum or minimum, we need the first derivative of the function. The derivative is given by \( y' = \frac{d}{dx}(ax^2 + bx + c) = 2ax + b \).

Step 2: Set the derivative to zero

To find the critical points where the function could have a maximum or minimum, set the first derivative \( y' = 2ax + b \) to zero. Thus, \( 2ax + b = 0 \).

Step 3: Solve for x

Rearrange the equation \( 2ax + b = 0 \) to solve for \( x \). This gives \( 2ax = -b \). Divide both sides by \( 2a \) to get \( x = -\frac{b}{2a} \).

Step 4: Verify the maximum or minimum

The value \( x = -\frac{b}{2a} \) gives the position of a maximum or minimum of the quadratic function. This is the vertex of the parabola, and the nature can be ascertained by checking the second derivative, \( y'' = 2a \). If \( a > 0 \), it's a minimum, and if \( a < 0 \), it's a maximum.

Key Concepts

Calculus

Calculus is an essential field of mathematics that helps us understand how things change. It mainly deals with two core ideas: differentiation and integration. Differentiation involves finding the rate of change of a quantity, while integration is about the accumulation of quantities. In simple terms, calculus allows us to investigate how functions behave as variables within them change.

The basic element of calculus, the function, is akin to a machine that transforms input values into outputs. With calculus, we can analyze these transformations and understand the behavior of functions in detail. Whether it's understanding motion, growth, or any natural process, calculus provides the tools to explore and predict changes.

Derivative

Calculating the derivative of a function helps us find how the output of that function changes as its input shifts. In other words, it tells us the function's rate of change at any given point. For a given quadratic function like \( y = ax^2 + bx + c \), the derivative is a linear expression: \( y' = 2ax + b \).

The derivative formula gives us the slope of the tangent line to the curve at any point. When we set the derivative equal to zero, we find the points where the slope of the tangent line is zero. These are known as critical points, where the function can have a maximum, a minimum, or a point of inflection. By understanding derivatives, we can make predictions about the original function's behavior.

Maximum and Minimum Points

The goal of finding maximum and minimum points of a function is to understand where the function reaches its highest or lowest values, respectively. For quadratic functions, these are usually referred to as the vertex of the parabola.

To find these points, start by taking the derivative of the function. For \( y = ax^2 + bx + c \) which yields \( y' = 2ax + b \). Set the derivative equal to zero to find the critical points, leading us to \( x = -\frac{b}{2a} \).

By determining the second derivative, \( y'' = 2a \), we can confirm whether the critical point is a maximum or minimum. If \( a > 0 \), the parabola opens upwards, indicating a minimum point at the vertex. Conversely, if \( a < 0 \), the parabola opens downwards, signifying a maximum point at the vertex.

Quadratic Equations

Quadratic equations are mathematical expressions involving a variable squared (\( x^2 \)). They take the general form: \( y = ax^2 + bx + c \), where \( a, b, \) and \( c \) are constants. The graph of a quadratic equation is a parabola, which is U-shaped for \( a > 0 \) or inverted U-shaped for \( a < 0 \).

Quadratics have crucial properties, such as the vertex and the axis of symmetry. The axis of symmetry is a vertical line passing through the vertex providing balance to the parabola. To solve quadratic equations, you may need to employ various methods like factoring, using the quadratic formula, or completing the square. Understanding the properties of quadratic equations allows us not just to solve for specific values but also to predict and describe the motion or shape defined by the parabolic curve.