Problem 28 - University Calculus: Early Transcendentals

Question

Find an equation of the straight line having slope \(1 / 4\) that is tangent to the curve \(y=\sqrt{x}\).

Step-by-Step Solution

Answer

The equation of the tangent line is \( y = \frac{1}{4}x + 1 \).

Step 1: Write down the equation of the line

The equation of a line with slope \( m = \frac{1}{4} \) is \( y = \frac{1}{4}x + c \), where \( c \) is the y-intercept. Our goal is to find \( c \).

Step 2: Differentiate the curve

To find the point of tangency, differentiate the curve \( y = \sqrt{x} \). The derivative is \( y' = \frac{d}{dx}(x^{1/2}) = \frac{1}{2\sqrt{x}} \). This gives the slope of the tangent to the curve at any point \( x \).

Step 3: Set the derivative equal to the line's slope

Set the derived slope equation \( \frac{1}{2\sqrt{x}} \) equal to the slope of the line \( \frac{1}{4} \) to find the x-coordinate of the point of tangency. Solve \( \frac{1}{2\sqrt{x}} = \frac{1}{4} \).

Step 4: Solve for the x-coordinate

From \( \frac{1}{2\sqrt{x}} = \frac{1}{4} \), multiply both sides by \( 2\sqrt{x} \) to get \( 1 = \frac{\sqrt{x}}{2} \). Multiply both sides by \( 2 \) to obtain \( \sqrt{x} = 2 \). Squaring both sides gives \( x = 4 \).

Step 5: Find the y-coordinate

Substitute \( x = 4 \) into the curve equation \( y = \sqrt{x} \) to find the y-coordinate. Thus \( y = \sqrt{4} = 2 \). The point of tangency is \( (4, 2) \).

Step 6: Find the y-intercept c of the tangent line

Substitute the x-value, y-value, and slope into the line equation \( y = \frac{1}{4}x + c \). Therefore: \( 2 = \frac{1}{4}(4) + c \). Simplifying gives \( 2 = 1 + c \), so \( c = 1 \).

Step 7: Write the final equation of the tangent line

With slope \( \frac{1}{4} \) and y-intercept \( c = 1 \), the equation of the tangent line is \( y = \frac{1}{4}x + 1 \).

Key Concepts

Tangent Line

A tangent line is a straight line that touches a curve at exactly one point without crossing it. This special line shares the same slope as the curve at the point of contact. It provides a linear approximation of the curve near this point.
To find a tangent line, you generally follow these steps:

• Identify the function or curve you are dealing with, such as \( y = \sqrt{x} \).
• Calculate the derivative to find the slope of the tangent at any point on the curve.
• Use the slope and a point on the curve to write the equation of the tangent line.

In this problem, you are given the slope of the tangent line directly. You find the point on the curve where the slope matches the given one, allowing you to write the tangent's equation easily.

Derivative

The derivative of a function measures how the function value changes as its input changes. In simple terms, it gives the slope of the function at any given point.
For the curve \( y = \sqrt{x} \), the derivative is calculated as: \[ y' = \frac{d}{dx}(x^{1/2}) = \frac{1}{2\sqrt{x}}. \]

• This derivative tells us the rate at which \( y \) is changing with respect to \( x \).
• To find where a line is tangent to the curve, set the derivative equal to the desired slope.

Calculating the derivative is crucial because it allows us to find the exact point on the curve that has a specified slope.

Slope

The slope of a line is a measure of its steepness, usually given as \( m \) in line equations. The slope represents how much \( y \) changes for a unit change in \( x \). In equations like \( y = mx + c \), \( m \) is the slope.
In this exercise, the slope of the tangent line is \( \frac{1}{4} \).

• To find the tangent point on \( y = \sqrt{x} \), you set the derivative equal to this value.
• Solve the equation \( \frac{1}{2\sqrt{x}} = \frac{1}{4} \) to find the x-coordinate where the curve's slope matches \( \frac{1}{4} \).

This step is necessary to locate the point on the curve where the tangent line just touches it.

Curve

A curve is a line that bends continuously without any sharp turns. In mathematics, curves like \( y = \sqrt{x} \) are graphical representations of a function.

• The curve \( y = \sqrt{x} \) rises slowly as \( x \) increases, resulting in less steep curves.
• To talk tangents, the derivative is used to analyze how the curve behaves at any specific point.

Understanding the shape and behavior of the curve helps in recognizing how and where tangent lines can be drawn with particular slopes. This case shows how a line with a small slope \( \frac{1}{4} \) just gently touches the curve at a single point.