Problem 15 - Calculus

Question

Choose \(c\) so that \(y=4 x\) is tangent to \(y=x^{2}+c .\) Match heights as well as slopes.

Step-by-Step Solution

Answer

Choose \(c = 4\).

Step 1: Set the Equations Equal to Find Intersection

To find the point where the line is tangent to the curve, set the equations equal: \(4x = x^2 + c\). Solve for \(x\) to find potential intersection points. Rearrange to form a quadratic: \(x^2 - 4x + c = 0\).

Step 2: Find Discriminant for Tangency Condition

For tangency, the quadratic equation should have exactly one solution. Thus, the discriminant must be zero. The discriminant \(\Delta\) is given by \(b^2 - 4ac\). For our equation: \((-4)^2 - 4 \times 1 \times c = 0\), simplifying to \(16 - 4c = 0\).

Step 3: Solve Discriminant Equation for \(c\)

Solve \(16 - 4c = 0\) for \(c:\) \[4c = 16 \Rightarrow c = 4.\] We find that \(c = 4\).

Step 4: Verify with Slope Condition

The derivative of \(y = x^2 + c\) gives the slope \(\frac{dy}{dx} = 2x\). For tangency, set the slope equal to the slope of the line (4): \(2x = 4\), solving to find \(x = 2\). Verify that substituting \(x=2\) in both equations satisfies height: \(y = 4 \times 2 = 8\) and \(y = 2^2 + 4 = 4 + 4 = 8\). Both conditions match.

Key Concepts

Understanding Quadratic Equations

Quadratic equations are a class of polynomial equations where the highest power of the unknown variable is squared. The general form is usually written as \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(x\) represents the variable we are solving for.
Quadratic equations are essential in a variety of mathematical contexts, particularly in finding where two curves intersect. The solutions to these equations tell us the \(x\)-values where these intersections occur.
To solve a quadratic equation, we often use methods like:

• Factoring: This involves expressing the quadratic as a product of two binomials.
• Using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) gives the roots directly.
• Completing the square: This method involves rewriting the quadratic in a form that reveals the vertex of the parabola.

Each method has its own advantages depending on the particular form of the equation. Understanding these methods helps in determining not only where two functions intersect but also how they behave around those intersection points.

Exploring the Discriminant Condition

A unique feature of quadratic equations is their discriminant, denoted by \(b^2 - 4ac\). The discriminant reveals critical characteristics about the roots of the quadratic equation. More importantly, it helps us understand the nature of the intersections between quadratic curves and other lines.
Here's what the discriminant tells us:

• If the discriminant is positive (\(b^2 - 4ac > 0\)), the quadratic equation has two distinct real roots, indicating two intersection points.
• If it is zero (\(b^2 - 4ac = 0\)), the equation has exactly one real root, meaning the line is tangent to the curve at this point. This is essential for identifying tangency.
• If the discriminant is negative (\(b^2 - 4ac < 0\)), there are no real roots, thus no real intersections.

For our specific problem, ensuring that the discriminant is zero is crucial because it signifies that the line only touches the parabola at one point, demonstrating perfect tangency.

Insight into Derivatives and Slopes

In calculus, derivatives are a powerful tool that provide information about the slope of a curve at any given point. In simple terms, they tell us how steep a line is at that particular location.
The derivative of a function is found by applying rules of differentiation, a process that measures the instantaneous rate of change of the function.
Let's take the quadratic function \(y = x^2 + c\):

• The derivative, \(\frac{dy}{dx} = 2x\), gives us the slope of the tangent line to the curve at any point \(x\).

For tangency between a line and a curve, their slopes must match at the point of contact. This means we set the derivative equal to the slope of the line. For example, if the line \(y = 4x\) is tangent to the curve \(y = x^2 + c\), we need \(2x = 4\), leading to \(x = 2\).
By ensuring that the slope conditions are satisfied, we verify that the two curves touch at the same point with matching slopes, confirming tangency.