Problem 51 - Thomas Calculus

Question

Tangent to a parabola Does the parabola \(y=2 x^{2}-13 x+5\) have a tangent whose slope is \(-1 ?\) If so, find an equation for the line and the point of tangency. If not, why not?

Step-by-Step Solution

Answer

Yes, the parabola has a tangent with slope -1 at (3, -16): the line is \(y = -x - 13\).

Step 1: Differentiate the Parabola

To find the slope of the tangent line to the parabola, we first need to differentiate the given equation. The equation is \( y = 2x^2 - 13x + 5 \). Using basic differentiation rules, the derivative is \( \frac{dy}{dx} = 4x - 13 \).

Step 2: Set the Derivative Equal to the Given Slope

The problem asks whether there is a tangent with a slope of \(-1\). We set \( 4x - 13 = -1 \) to find the \(x\) values where the parabola has this slope.

Step 3: Solve for x

Solve the equation \( 4x - 13 = -1 \). Add 13 to both sides to get \( 4x = 12 \). Then, divide by 4 to solve for \( x \), which gives \( x = 3 \).

Step 4: Find the Corresponding y-Value

Substitute \( x = 3 \) back into the original parabola equation to find the \(y\)-coordinate of the point of tangency:\( y = 2(3)^2 - 13(3) + 5 \)}, which simplifies to \( y = 18 - 39 + 5 \), resulting in \( y=-16 \). Thus, the point of tangency is \( (3, -16) \).

Step 5: Write the Equation of the Tangent Line

Using the point-slope form of a line, \( y - y_1 = m(x - x_1) \), and using the point \((3, -16)\) and the slope \(-1\), we have:\( y + 16 = -1(x - 3) \).Simplifying, we obtain the equation of the tangent line: \( y = -x - 13 \).

Key Concepts

Parabola

The term "parabola" might sound fancy, but it's a mathematical shape you encounter often. Imagine throwing a ball. As it travels through the air, it makes a gentle arc. That arc, most of the time, is a parabola.

A parabola is a symmetric curve shaped like an open bowl. It's described by a quadratic equation of the form \( y = ax^2 + bx + c \). Here, \( a \), \( b \), and \( c \) are constants that determine how the parabola looks or is positioned.

• "Vertex" is the peak or lowest point of the parabola.
• "Axis of symmetry" is a vertical line that runs through the vertex, dividing the parabola into mirror images.
• "Focus" is a point inside the parabola where rays of light parallel to the axis reflect.

Understanding these basic components can help you visualize how a parabola behaves and why it's important in various fields like physics and engineering. In our exercise, the parabola is given by the equation \( y = 2x^2 - 13x + 5 \). This tells us how the curve bends and where it sits on a graph.

Differentiation

Differentiation is a handy mathematical tool used to find the rate at which something changes. It's like finding out how fast a car is going at any given moment, given its movement data.

In calculus, differentiation applies to functions, and we typically talk about finding a derivative. The derivative represents the slope of the tangent to a curve at any point. For a given function, say, \( y = 2x^2 - 13x + 5 \), the process of differentiation gives us \( \frac{dy}{dx} \), which is the rate of change of \( y \) with respect to \( x \).

In our exercise, the derivative of the parabola is \( \frac{dy}{dx} = 4x - 13 \). This derivative allows us to find the slope of the tangent line at any point on the parabola.

• "Use differentiation to determine instantaneous change." For example, what is the slope of the tangent exactly where you are?
• "The derivative tells you whether the function increases or decreases as you move along the curve."

This fundamental concept is crucial in many real-world applications, including physics, engineering, and economics, where understanding the change is essential.

Slope of a line

The slope of a line is a fundamental concept in mathematics that tells you how steep the line is. It's like figuring out how steep a hill is when you're hiking.

The slope is often represented by the letter \( m \) and is calculated as the rise over the run, or \( \frac{\Delta y}{\Delta x} \). In simpler terms, it's how much the line goes up or down for each unit it moves sideways.

• If \( m > 0 \), the line is rising to the right.
• If \( m < 0 \), the line is falling to the right.
• If \( m = 0 \), the line is flat, or horizontal.

In the context of our exercise, when we talk about the tangent line to the parabola \( y = 2x^2 - 13x + 5 \) having a slope, we're seeing how the curve's direction changes at a specific point. With a slope of \( -1 \), the tangent line runs downhill, and this particular slope helps in forming the equation of the tangent using the point-slope formula. Understanding slope not only helps in geometry but is also vital in calculus and analytics.