2.2 Tangent Lines And The Derivative Homework Answer Key

2.2 Tangent Lines and the Derivative: Homework Answer Key & Deep Dive

This comprehensive guide tackles the complexities of Section 2.2, focusing on tangent lines and the derivative. We'll move beyond simply providing answers; we'll delve into the underlying concepts, offering explanations, examples, and strategies to master this crucial calculus topic. Understanding tangent lines and their relationship to the derivative is fundamental to grasping the power and applications of calculus.

Understanding Tangent Lines

Before we jump into the homework solutions, let's solidify our understanding of tangent lines.

What is a Tangent Line? A tangent line touches a curve at exactly one point, sharing the same instantaneous rate of change (slope) as the curve at that point. Imagine a car driving along a curvy road; the tangent line represents the direction the car is heading at a specific moment.

Key Concepts:

• Secant Line: A line that intersects a curve at two or more points. The slope of a secant line represents the average rate of change between those points.
• Limit: As the two points on the curve get infinitely close, the secant line approaches the tangent line, and its slope approaches the instantaneous rate of change. This concept is crucial in defining the derivative.
• Point of Tangency: The point where the tangent line touches the curve.

Finding the Equation of a Tangent Line:

To find the equation of a tangent line at a specific point (x₁, y₁) on a curve defined by a function f(x), we need two things:

1. The point (x₁, y₁): This is given or can be calculated by substituting the x-coordinate into the function: y₁ = f(x₁).
2. The slope (m): This is the derivative of the function evaluated at the point x₁, denoted as f'(x₁). The derivative represents the instantaneous rate of change of the function at that point.

Once we have the point and the slope, we can use the point-slope form of a line: y - y₁ = m(x - x₁)

The Derivative: A Formal Definition

The derivative is a fundamental concept in calculus. It quantifies the instantaneous rate of change of a function at a specific point. Formally, the derivative of a function f(x) at a point x is defined as the limit:

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

This limit represents the slope of the tangent line to the curve of f(x) at the point x. If this limit exists, the function is said to be differentiable at x.

Interpreting the Derivative

The derivative has several key interpretations:

• Instantaneous Rate of Change: As mentioned, it tells us how fast the function is changing at a precise moment. This is crucial in various applications, such as calculating velocity (the derivative of position) or acceleration (the derivative of velocity).
• Slope of the Tangent Line: Geometrically, it represents the slope of the tangent line to the curve at a given point.
• Approximation: The derivative can be used to approximate the value of the function near a given point using linear approximation (tangent line approximation).

Homework Problems & Solutions: A Detailed Walkthrough

Let's tackle some typical homework problems from Section 2.2. Since we don't have access to your specific assignment, I'll provide examples covering a range of difficulty levels. Remember to always refer to your textbook and lecture notes for specific problem types and notations.

Problem 1: Finding the Equation of a Tangent Line

Find the equation of the tangent line to the curve y = x² + 2x - 3 at the point x = 1.

Solution:

1. Find the point: When x = 1, y = (1)² + 2(1) - 3 = 0. So the point is (1, 0).
2. Find the derivative: f(x) = x² + 2x - 3. Using the power rule, f'(x) = 2x + 2.
3. Find the slope at x = 1: f'(1) = 2(1) + 2 = 4. The slope is 4.
4. Use the point-slope form: y - 0 = 4(x - 1). Simplifying, we get y = 4x - 4.

Therefore, the equation of the tangent line is y = 4x - 4.

Problem 2: Finding the Derivative Using the Limit Definition

Find the derivative of f(x) = 3x² - 2x + 1 using the limit definition.

Solution:

We use the limit definition: f'(x) = lim (h→0) [(f(x + h) - f(x)) / h]

1. Find f(x + h): f(x + h) = 3(x + h)² - 2(x + h) + 1 = 3(x² + 2xh + h²) - 2x - 2h + 1 = 3x² + 6xh + 3h² - 2x - 2h + 1
2. Substitute into the limit: f'(x) = lim (h→0) [(3x² + 6xh + 3h² - 2x - 2h + 1 - (3x² - 2x + 1)) / h]
3. Simplify: f'(x) = lim (h→0) [(6xh + 3h² - 2h) / h] = lim (h→0) [6x + 3h - 2]
4. Evaluate the limit: As h approaches 0, the expression simplifies to 6x - 2.

Therefore, the derivative of f(x) = 3x² - 2x + 1 is f'(x) = 6x - 2.

Problem 3: Applications of the Derivative

A ball is thrown upward with an initial velocity of 64 ft/s. Its height (in feet) after t seconds is given by h(t) = -16t² + 64t. Find the velocity of the ball at t = 2 seconds.

Solution:

1. Find the derivative: The velocity is the derivative of the height function. h'(t) = -32t + 64.
2. Evaluate at t = 2: h'(2) = -32(2) + 64 = 0.

The velocity of the ball at t = 2 seconds is 0 ft/s. This means the ball has reached its maximum height and is momentarily stationary before falling back down.

Advanced Topics and Extensions

This section provides a glimpse into more advanced concepts related to tangent lines and derivatives.

Higher-Order Derivatives: The derivative of a derivative is called the second derivative, denoted as f''(x). Similarly, we can find third, fourth, and higher-order derivatives. These higher-order derivatives provide information about the concavity and other properties of the function.

Implicit Differentiation: This technique is used to find the derivative of functions that are not explicitly defined as y = f(x). It involves differentiating both sides of the equation with respect to x and then solving for dy/dx.

Related Rates: This application involves finding the rate of change of one quantity in terms of the rate of change of another quantity. These problems often involve implicit differentiation and require careful analysis of the relationships between variables.

Optimization Problems: The derivative is a powerful tool for solving optimization problems, which involve finding the maximum or minimum values of a function. Setting the derivative equal to zero and analyzing the second derivative helps determine these extreme values.

Conclusion: Mastering Tangent Lines and the Derivative

This comprehensive guide has explored the fundamental concepts of tangent lines and the derivative, providing detailed solutions to example problems and highlighting essential applications. Remember that consistent practice is key to mastering these concepts. Work through various problems, focusing on understanding the underlying principles, not just memorizing formulas. By understanding the connection between tangent lines, the limit definition of the derivative, and its various interpretations, you'll build a solid foundation for your calculus journey. The power of calculus lies in its ability to model and solve real-world problems, and a strong grasp of these foundational concepts is essential for success in more advanced topics.