2.2 Tangent Lines And The Derivative Homework Answer Key

2.2 Tangent Lines and the Derivative: A Comprehensive Guide with Solved Problems

Calculus, a cornerstone of higher mathematics, introduces many fundamental concepts. Among these, the tangent line and its close relative, the derivative, hold significant importance. Understanding how to find the equation of a tangent line at a specific point on a curve is crucial for grasping the essence of calculus and its numerous applications in various fields, from physics and engineering to economics and computer science. This article delves deep into section 2.2, focusing on tangent lines and the derivative, providing a comprehensive explanation with numerous solved problems to solidify your understanding.

Understanding Tangent Lines

A tangent line, in its simplest form, is a line that touches a curve at a single point without crossing it. Imagine a circle; a tangent line would graze the circle at only one point. For more complex curves, the concept remains similar. The key is that the tangent line shares the same instantaneous rate of change as the curve at the point of tangency. This instantaneous rate of change is precisely what the derivative helps us calculate.

Key Properties of Tangent Lines:

• Single Point of Contact: A tangent line only touches the curve at one point in a sufficiently small neighborhood.
• Instantaneous Rate of Change: The slope of the tangent line represents the instantaneous rate of change of the function at the point of tangency.
• Approximation: The tangent line provides a good linear approximation of the curve near the point of tangency.

The Derivative: Measuring Instantaneous Change

The derivative is a powerful tool that allows us to find the instantaneous rate of change of a function at any point. It's the slope of the tangent line to the curve at that specific point. The derivative of a function f(x), denoted as f'(x) or df/dx, represents this instantaneous rate of change.

Finding the Derivative:

The derivative is typically found using the limit definition:

f'(x) = lim (h→0) [(f(x + h) - f(x)) / h]

This formula represents the slope of the secant line connecting two points on the curve as the distance between those points approaches zero. The limit of this slope as the distance approaches zero gives us the slope of the tangent line, hence the derivative.

Example 1: Finding the Derivative

Let's find the derivative of the function f(x) = x².

Using the limit definition:

f'(x) = lim (h→0) [((x + h)² - x²) / h] = lim (h→0) [(x² + 2xh + h² - x²) / h] = lim (h→0) [(2xh + h²) / h] = lim (h→0) [2x + h] = 2x

Therefore, the derivative of f(x) = x² is f'(x) = 2x.

Finding the Equation of a Tangent Line

Once we have the derivative (which gives us the slope of the tangent line), finding the equation of the tangent line at a specific point becomes straightforward. We use the point-slope form of a line:

y - y₁ = m(x - x₁)

where:

• y and x are the variables
• y₁ and x₁ are the coordinates of the point of tangency
• m is the slope of the tangent line (which is the derivative evaluated at x₁)

Example 2: Finding the Equation of a Tangent Line

Let's find the equation of the tangent line to the curve f(x) = x² at the point (2, 4).

1. Find the derivative: From Example 1, we know that f'(x) = 2x.

2. Find the slope at the point of tangency: Substitute x = 2 into the derivative: f'(2) = 2(2) = 4. This is the slope (m) of the tangent line.

3. Use the point-slope form: The point of tangency is (2, 4), so x₁ = 2 and y₁ = 4.

y - 4 = 4(x - 2) y - 4 = 4x - 8 y = 4x - 4

Therefore, the equation of the tangent line to f(x) = x² at (2, 4) is y = 4x - 4.

Advanced Applications and Problem Solving

The concepts of tangent lines and derivatives extend far beyond simple polynomial functions. Let's explore some more complex scenarios and solved problems:

Example 3: Tangent Line to a Trigonometric Function

Find the equation of the tangent line to the curve f(x) = sin(x) at x = π/2.

1. Find the derivative: f'(x) = cos(x)

2. Find the slope at x = π/2: f'(π/2) = cos(π/2) = 0

3. Find the y-coordinate at x = π/2: f(π/2) = sin(π/2) = 1

4. Use the point-slope form: The point is (π/2, 1), and the slope is 0.

y - 1 = 0(x - π/2) y = 1

The equation of the tangent line is y = 1. This is a horizontal line, indicating that the function has a zero instantaneous rate of change at x = π/2.

Example 4: Finding Points with Horizontal Tangent Lines

Find the x-coordinates where the tangent line to f(x) = x³ - 3x + 2 is horizontal.

1. Find the derivative: f'(x) = 3x² - 3

2. Set the derivative equal to zero: Horizontal tangent lines have a slope of zero. Therefore, we solve 3x² - 3 = 0.

3x² = 3 x² = 1 x = ±1

The tangent lines are horizontal at x = 1 and x = -1.

Example 5: Tangent Line to a Rational Function

Find the equation of the tangent line to f(x) = 1/x at x = 2.

1. Find the derivative: Using the power rule or the quotient rule, we find f'(x) = -1/x².

2. Find the slope at x = 2: f'(2) = -1/2² = -1/4

3. Find the y-coordinate at x = 2: f(2) = 1/2

4. Use the point-slope form: The point is (2, 1/2), and the slope is -1/4.

y - 1/2 = (-1/4)(x - 2) y = (-1/4)x + 1

Applications in Real-World Scenarios

The concepts of tangent lines and derivatives are fundamental to understanding and modeling numerous real-world phenomena:

• Physics: Velocity is the derivative of position with respect to time. Acceleration is the derivative of velocity with respect to time. Tangent lines help visualize instantaneous velocity at a specific moment.

• Engineering: Optimizing designs often involves finding maximum or minimum values of a function, which requires finding critical points where the derivative is zero.

• Economics: Marginal cost, marginal revenue, and marginal profit are all derivatives that describe the instantaneous rate of change of cost, revenue, and profit, respectively.

• Computer Science: Numerical methods for solving equations often use the derivative to approximate solutions iteratively.

This comprehensive guide provides a robust foundation in understanding tangent lines and the derivative, covering essential concepts and offering diverse solved examples to solidify your grasp of the subject. Remember, practice is key! Work through additional problems to build your confidence and proficiency in applying these crucial calculus principles. Further exploration of different types of functions and derivative rules will enhance your understanding even more.