2018 Methods Exam 1 Solutions

2018 Methods Exam 1 Solutions: A Comprehensive Guide

This article provides a detailed walkthrough of the solutions for the 2018 Methods Exam 1. We'll cover each question thoroughly, explaining the concepts involved and demonstrating the necessary steps for solving each problem. This guide is intended for students studying mathematical methods, offering a valuable resource for exam revision and a deeper understanding of key mathematical concepts. Understanding these solutions is crucial for success in further mathematical studies. We will focus on clarity and step-by-step explanations to help solidify your understanding.

Understanding the Exam Context

Before diving into the solutions, let's briefly recall the typical structure and content of a Methods Exam 1. These exams usually assess foundational knowledge in calculus, including differentiation, integration, and applications of both. They often feature questions on:

• Functions and their graphs: Analyzing domains, ranges, asymptotes, and transformations.
• Differentiation: Finding derivatives using various rules (product, quotient, chain rule), applications to finding tangents and normals, rates of change, and optimization problems.
• Integration: Evaluating definite and indefinite integrals using various techniques, applications to finding areas under curves and volumes of revolution.
• Applications of calculus: Solving real-world problems using calculus, such as optimization problems (maximizing profit, minimizing cost) and related rates problems.

Detailed Solutions: A Step-by-Step Approach

Note: Since the specific questions from the 2018 Methods Exam 1 are not provided, I will create example problems representative of the typical questions found in such an exam. These examples will cover the key concepts and techniques you would encounter.

Example Problem 1: Differentiation and Tangent Lines

Question: Find the equation of the tangent line to the curve y = x³ - 6x + 5 at the point where x = 2.

Solution:

1. Find the derivative: We first need to find the derivative of the function y = x³ - 6x + 5. Using the power rule of differentiation, we get: dy/dx = 3x² - 6

2. Find the slope at x = 2: Substitute x = 2 into the derivative to find the slope of the tangent line at that point: dy/dx|_(x=2) = 3(2)² - 6 = 12 - 6 = 6

3. Find the y-coordinate: Substitute x = 2 into the original function to find the y-coordinate of the point: y = (2)³ - 6(2) + 5 = 8 - 12 + 5 = 1

4. Write the equation of the tangent line: Now we use the point-slope form of a line, y - y₁ = m(x - x₁), where m is the slope and (x₁, y₁) is the point. In this case, m = 6 and (x₁, y₁) = (2, 1). Therefore, the equation of the tangent line is: y - 1 = 6(x - 2) y = 6x - 11

Example Problem 2: Optimization Problem

Question: A rectangular garden is to be enclosed by a fence on three sides and a wall on the fourth side. If 50 meters of fencing is available, what dimensions will maximize the area of the garden?

Solution:

1. Define variables: Let x be the length of the side parallel to the wall, and y be the length of each of the other two sides.

2. Write the constraint equation: The total length of the fencing is 50 meters, so we have: x + 2y = 50

3. Write the objective function: We want to maximize the area of the garden, which is given by A = xy.

4. Express one variable in terms of the other: Solve the constraint equation for one variable (let's solve for x): x = 50 - 2y

5. Substitute into the objective function: Substitute the expression for x into the area equation: A(y) = (50 - 2y)y = 50y - 2y²

6. Find the critical points: To find the maximum area, we take the derivative of A(y) with respect to y and set it to zero: dA/dy = 50 - 4y = 0 y = 12.5

7. Check for maximum: We use the second derivative test to confirm that this is a maximum: d²A/dy² = -4 Since the second derivative is negative, we have a maximum.

8. Find the dimensions: Substitute y = 12.5 back into the equation for x: x = 50 - 2(12.5) = 25

9. State the answer: The dimensions that maximize the area are 25 meters by 12.5 meters.

Example Problem 3: Integration and Area Under a Curve

Question: Find the area enclosed by the curve y = x² - 4x + 5, the x-axis, and the lines x = 1 and x = 3.

Solution:

1. Set up the definite integral: The area under the curve is given by the definite integral: Area = ∫[1, 3] (x² - 4x + 5) dx

2. Evaluate the integral: Using the power rule for integration, we get: Area = [x³/3 - 2x² + 5x] evaluated from x = 1 to x = 3 Area = [(3³/3 - 2(3)² + 5(3)) - (1³/3 - 2(1)² + 5(1))] = (9 - 18 + 15) - (1/3 - 2 + 5) = 6 - 10/3 = 8/3

Therefore, the area enclosed is 8/3 square units.

Further Topics and Considerations

The examples above illustrate fundamental concepts tested in a typical Methods Exam 1. However, exams may also include more advanced topics such as:

• Implicit Differentiation: Finding derivatives of functions defined implicitly.
• Related Rates Problems: Problems involving rates of change of related variables.
• Volumes of Revolution: Finding volumes of solids generated by revolving curves around axes.
• Applications of Integration: Solving problems involving work, average value, and more.

Frequently Asked Questions (FAQ)

Q1: What resources are available to help me prepare for the Methods Exam 1?

A1: Beyond this guide, consider utilizing textbooks, past papers, online resources, and practice exercises. Working through numerous problems is key to mastering the concepts.

Q2: What are some common mistakes students make on this exam?

A2: Common mistakes include: incorrectly applying differentiation rules, making errors in integration, and failing to properly interpret problem statements. Careful attention to detail is crucial.

Q3: How can I improve my problem-solving skills in mathematical methods?

A3: Practice is key. Regularly solve problems of varying difficulty, focusing on understanding the underlying concepts rather than just memorizing formulas.

Conclusion

This comprehensive guide provides a solid foundation for understanding the solutions to a typical Methods Exam 1. Remember that consistent practice and a thorough understanding of the underlying concepts are essential for success. By working through numerous examples and seeking clarification when needed, you can confidently approach the exam and achieve your desired results. Remember that mathematics is a journey of understanding, and persistent effort will lead to mastery.