Problem 33 - Essential Calculus Early Transcendentals

Question

Find the unit vectors that are parallel to the tangent line to the parabola \(y=x^{2}\) at the point \((2,4)\)

Step-by-Step Solution

Answer

Unit vectors are \( \langle \frac{1}{\sqrt{17}}, \frac{4}{\sqrt{17}} \rangle \) and \( \langle -\frac{1}{\sqrt{17}}, -\frac{4}{\sqrt{17}} \rangle \).

Step 1: Differentiate the Parabola

To find the slope of the tangent line to the parabola at a given point, differentiate the equation of the parabola with respect to x. The given parabola is \( y = x^2 \). Thus, the derivative is: \( \frac{dy}{dx} = 2x \).

Step 2: Find the Slope at the Point

Substitute the x-coordinate of the given point into the derivative to find the slope of the tangent line at that point. Here, \( x = 2 \), so the slope \( m \) at point \( (2, 4) \) is: \( m = 2 \times 2 = 4 \).

Step 3: Form the Direction Vector of the Tangent Line

A direction vector for the tangent line at \( (2, 4) \) based on the slope \( m = 4 \) can be expressed as \( \langle 1, 4 \rangle \). This indicates that for each unit increase in x, y increases by 4 units.

Step 4: Normalize the Direction Vector

To find the unit vector, compute the magnitude of the vector \( \langle 1, 4 \rangle \) using the formula \( \text{magnitude} = \sqrt{1^2 + 4^2} = \sqrt{17} \). The unit vector in the direction of \( \langle 1, 4 \rangle \) is then \( \frac{1}{\sqrt{17}} \langle 1, 4 \rangle \), which simplifies to \( \langle \frac{1}{\sqrt{17}}, \frac{4}{\sqrt{17}} \rangle \).

Step 5: Find the Opposite Unit Vector

The opposite unit vector, which is also parallel to the tangent line, is simply the negative of the unit vector. That is: \( \langle -\frac{1}{\sqrt{17}}, -\frac{4}{\sqrt{17}} \rangle \).

Key Concepts

Derivative

In calculus, a derivative is a powerful concept that helps us understand how a function changes at any given point. The derivative of a function provides the slope of the tangent line to the curve at that point. For the function \( y = x^2 \) representing a parabola, the derivative is determined by using differentiation rules. When you differentiate \( y = x^2 \), you obtain \( \frac{dy}{dx} = 2x \). This result tells us how steep the curve is at any point \( x \). For instance, if \( x = 2 \), the slope will be \( 2 \times 2 = 4 \).
Understanding derivatives is invaluable as they offer insights into velocity, acceleration, and even rates of change in various fields like physics, economics, and biology.

Tangent Line

A tangent line is a straight line that touches a curve at precisely one point, without crossing it. It's significant because at the point of contact, the tangent line has the same direction as the curve. To find this line for a curve like a parabola, you use the derivative.

• With the parabola \( y = x^2 \), and its derivative \( \frac{dy}{dx} = 2x \), the slope of the tangent at \( x = 2 \) is 4.
• The line with this slope that touches the point \((2, 4)\) is the tangent line.

This tangent line can be used for approximations or to understand the instantaneous rate of change, such as understanding the speed of a moving object at a specific moment.

Unit Vector

A unit vector is a vector that has a length of one unit but points in the same direction as the given vector. It helps in simplifying vector calculations by focusing on direction rather than magnitude.
To create a unit vector from a direction vector like \( \langle 1, 4 \rangle \):

• First, calculate its magnitude. For our vector, it is \( \sqrt{1^2 + 4^2} = \sqrt{17} \).
• Next, divide each component of the vector by its magnitude, yielding \( \langle \frac{1}{\sqrt{17}}, \frac{4}{\sqrt{17}} \rangle \).

This unit vector maintains the direction of the original vector and is often used in physics and engineering for defining directions.

Parabola

A parabola is a curve where any point is at an equal distance from a fixed point, called the focus, and a fixed line, called the directrix. In algebra, parabolas are typically expressed as a quadratic function like \( y = x^2 \).
For the parabola in our problem, \( y = x^2 \), its simplicity lies in its symmetric shape around the vertical axis.

• The vertex of this parabola is at the point \( (0,0) \), which is the lowest point on the curve.
• Parabolas often model various real-world phenomena like projectile motion or satellite dishes.

Grasping how a parabola works aids in understanding not just geometry but also how these curves appear naturally and how they can be utilized in technology.