Problem 41 - Thomas Calculus

Question

In Exercises \(41-50,\) determine all critical points for each function. $$ y=x^{2}-6 x+7 $$

Step-by-Step Solution

Answer

The critical point is \((3, -2)\).

Step 1: Identify the Function Components

The function provided is a quadratic function: \( y = x^2 - 6x + 7 \). We will identify the function's components to understand its structure and apply differentiation.

Step 2: Differentiate the Function

To find the critical points, first calculate the derivative of \( y \) with respect to \( x \). The derivative \( y' \) is determined using basic differentiation rules:\[y' = \frac{d}{dx}(x^2 - 6x + 7) = 2x - 6. \]This expression represents the slope of the tangent at any point \( x \) on the curve.

Step 3: Find Critical Points

A critical point occurs where the derivative \( y' \) is equal to zero or is undefined. In this case, solve the equation:\[2x - 6 = 0.\]Add 6 to both sides to get:\[2x = 6.\]Then, divide both sides by 2 to find:\[x = 3.\]

Step 4: Verify the Critical Point

Calculate \( y \) when \( x = 3 \) to find the corresponding \( y \)-value of the critical point:\[y = 3^2 - 6 \times 3 + 7 = 9 - 18 + 7 = -2.\]Thus, the critical point is \((3, -2)\).

Key Concepts

Quadratic Function

A quadratic function is a type of polynomial that is characterized by its degree of two. This means the highest power of the variable, typically represented by "x," is a square. In its standard form, a quadratic function is written as:

• \( y = ax^2 + bx + c \)

where \(a\), \(b\), and \(c\) are constants, with \(a eq 0\). This variable arrangement ensures we have a curve, commonly called a parabola.
Quadratic functions have several important features:

• The graph of a quadratic function is a symmetrical curve that can be either "U" shaped or "n" shaped, depending on the sign of "a." Positive "a" opens upwards, while negative "a" opens downwards.
• The vertex is the highest or lowest point of the parabola, representing either a maximum or a minimum value of the function.
• Quadratic functions will always intersect the y-axis at the point \((0, c)\), as this is the value of \(y\) when \(x = 0\).

Understanding quadratic functions is crucial as they model several real-world phenomena, from projectile motion to economics.

Derivative

The derivative of a function is a fundamental concept in calculus that measures how a function changes as its input changes. It gives us the rate of change or the "slope" of the function at any given point. Mathematically, the derivative of a function \( y = f(x) \) is denoted by \( y' \) or \( \frac{dy}{dx} \).
For polynomial functions like quadratics, differentiating is straightforward. Here are some basic rules:

• The derivative of \(x^n\) is \(nx^{n-1}\).
• Constants disappear, as their rate of change is zero.
• The derivative is additive, so you can differentiate each term separately and sum the results.

As an example, for our quadratic function \( y = x^2 - 6x + 7 \), its derivative is \( y' = 2x - 6 \). This derivative tells us how steep the graph of the function is at any point \(x\). It's a powerful tool used not only in math but in a plethora of scientific applications.

Differentiation

Differentiation is the process of finding the derivative of a function. The procedure provides a systematic way to determine the rate of change of one quantity with respect to another. It is a key operation in calculus and is used to analyze and predict behaviors within both natural and applied sciences.
The process of differentiation involves applying rules such as the power rule, product rule, quotient rule, and chain rule, depending on the type of function.

• The power rule is particularly handy for polynomials, as we've seen with our quadratic function.
• For the function \( y = x^2 - 6x + 7 \), using the power rule, we differentiate term by term to get the expression \( y' = 2x - 6 \).
• This helps determine crucial behaviors of the function, like identifying points where the slope is zero, which may indicate local maxima, minima, or points of inflection.

The ability to differentiate is foundational in higher mathematics, and understanding it is critical for studying more advanced topics.

Tangent Slope

The tangent slope of a curve at a given point is the slope of the line that just "touches" the curve at that point, matching the curve's direction. It reflects the instant rate of change of the curve at that exact spot and is synonymous with the derivative at that point.
For example, given a function \( y = f(x) \), its derivative \( f'(x) \) gives us the slope of the tangent at any point \( x \). With our quadratic function \( y = x^2 - 6x + 7 \), the derivative is \( y' = 2x - 6 \) which, when solved, helps identify such points.

• A zero slope (\( f'(x) = 0 \)) typically identifies critical points such as local maxima or minima.
• Finding \( x \) where this occurs determines the "flat" spot on the curve, where the direction of the slope changes.
• In our exercise, substituting \( x = 3 \) into the derivative gives us a slope of zero, confirming a critical point.

The concept of tangent slopes is crucial for optimization and analysis in many fields, as it provides vital information about the behavior of a function.