Problem 22 - Calculus: Early Transcendentals

Question

If the tangent line to \( y = f(x) \) at \( (4, 3) \) passes through the point \( (0, 2) \), find \( f(4) \) and \( f'(4) \).

Step-by-Step Solution

Answer

\( f(4) = 3 \) and \( f'(4) = \frac{1}{4} \).

Step 1: Identify the Given Information

We are given that the tangent line to the curve of the function \( y = f(x) \) at the point \( (4,3) \) passes through another point \( (0,2) \). This information suggests that the equation of the tangent line can be determined using the point-slope form of the line equation.

Step 2: Determine the Slope Using Two Points

We have the points \( (4,3) \) and \( (0,2) \). The slope \( m \) of the line through these two points is calculated as follows: \[ m = \frac{3 - 2}{4 - 0} = \frac{1}{4} \]

Step 3: Equation of the Tangent Line

Use the point-slope form of a line equation: \[ y - y_1 = m(x - x_1) \] where \((x_1, y_1)\) is a point on the line. We'll use \((4,3)\) as \((x_1, y_1)\) and \( m = \frac{1}{4} \).Thus, \[ y - 3 = \frac{1}{4}(x - 4) \] simplified to \[ y = \frac{1}{4}x + 2 \]

Step 4: Verify \( f(4) \) Using the Tangent Line

Since the tangent line at \( x=4 \) satisfies \( y = f(x) \), and we are given the point \( (4, 3) \) is on the curve, directly we have \( f(4) = 3 \).

Step 5: Determine \( f'(4) \) Using the Slope of Tangent Line

The slope of the tangent line represents \( f'(x) \) at the point \((x=4)\). Therefore, \( f'(4) = \frac{1}{4} \).

Key Concepts

Tangent Line

A tangent line is an important concept in calculus, especially when studying the properties of curves. Simply put, it is a straight line that touches a curve at a single point without crossing it in the immediate vicinity. The key aspect of the tangent line is that it provides an instantaneous rate of change or slope of the curve at that specific point. In the context of the given exercise, the tangent line to the function \( y = f(x) \) at \((4, 3)\) gives us information about how the curve behaves at that point.

Understanding tangent lines helps us approximate the function around the point of tangency. Essentially, the tangent line is the best linear approximation of the curve at that particular point. This makes it incredibly useful in various applications, from physics to engineering, where predicting changes is crucial.

When solving problems involving tangent lines:

• Identify the point of tangency
• Determine the slope of the tangent line
• Use the point-slope form to find the equation of the line

Derivative

The derivative is a fundamental concept in calculus, representing the rate at which a function is changing at any given point. In other words, it tells us how the function's output value changes as its input value changes. Mathematically, the derivative of a function \( f(x) \) at a point \( x=a \) is the slope of the tangent line to the curve at that point. This is why in our exercise, finding \( f'(4) \) involves the tangent line's slope.

The process of finding a derivative is known as differentiation. We often denote the derivative of \( f(x) \) as \( f'(x) \) or \( \frac{df}{dx} \). Understanding derivatives allows us to:

• Determine the slope of a curve at any point
• Predict how small changes in \( x \) affect changes in \( y \)
• Analyze real-world phenomena such as velocity and acceleration

With practice, calculating derivatives becomes an essential tool for solving complex problems involving rates of change.

Slope

Slope is a simple yet powerful concept, fundamental to understanding linear relationships. In geometry, the slope of a line is a measure of its steepness and direction. It's calculated as the ratio of the vertical change to the horizontal change between two points on the line. In mathematical terms, slope \( m \) is given by \( m = \frac{\Delta y}{\Delta x} \). In our exercise, we used the points \((4,3)\) and \((0,2)\) to find the slope of the tangent line, resulting in \( m = \frac{1}{4} \).

The slope communicates important information about how two variables relate. For instance:

• A positive slope indicates a positive correlation between variables, meaning as \( x \) increases, \( y \) does too.
• A negative slope indicates a negative correlation, where \( y \) decreases as \( x \) increases.
• A zero slope suggests a horizontal line, meaning no change in \( y \) despite changes in \( x \).

Understanding slope is key to analyzing linear equations, graphs, and real-world situations where modeling relationships is necessary.

Point-Slope Form

The point-slope form is a practical and widely used method for expressing the equation of a line. It is particularly useful when you have a point on the line and the slope. The formula is given by \( y - y_1 = m(x - x_1) \), where \((x_1, y_1)\) is a known point on the line and \( m \) is the slope. This formula makes it straightforward to derive the line's equation when solving calculus problems.

In our exercise, utilizing the point-slope form allowed us to find the tangent line's equation quickly once the slope \( \frac{1}{4} \) was determined. Here's how you can use it effectively:

• Identify a known point on the line
• Calculate the slope
• Plug these values into the point-slope form to find the line's equation

Mastering the point-slope form empowers students to handle a variety of problems involving linear equations and gives a deeper understanding of the relationships between points and slopes.