Pearson 3.6 Exercises Answers Calculus Cp

Mastering Calculus CP: A Deep Dive into Pearson 3.6 Exercises

Calculus is a fundamental branch of mathematics that deals with continuous change. From physics to economics, its applications are vast and ever-present. For students delving into Calculus CP (College Preparatory), Pearson's textbook is a common resource. Among the numerous exercises, section 3.6 often presents a significant challenge. This article provides a comprehensive guide to understanding and tackling the exercises in Pearson Calculus CP section 3.6, complete with explanations and strategies for success.

Understanding the Core Concepts of Section 3.6

Before diving into specific problems, it's crucial to grasp the underlying principles covered in Pearson Calculus CP section 3.6. Generally, this section focuses on related rates, a concept where the rate of change of one quantity is related to the rate of change of another. These problems often involve geometric shapes, physical scenarios, and the application of differentiation.

Key concepts to remember include:

• Implicit Differentiation: This technique allows us to find the derivative of a function where y is not explicitly defined in terms of x. Instead, we differentiate both sides of an equation with respect to a common variable (usually t for time) and then solve for the desired rate of change.
• Chain Rule: Essential for differentiating composite functions. Remember that d/dt [f(g(t))] = f'(g(t)) * g'(t).
• Geometric Formulas: Understanding the formulas for area, volume, and perimeter of common shapes is essential as many related rates problems involve these figures.
• Problem-Solving Strategy: A consistent approach is vital. This involves reading the problem carefully, drawing a diagram, identifying knowns and unknowns, establishing a relationship between the variables, differentiating, and finally, substituting values to solve.

A Step-by-Step Approach to Solving Related Rates Problems

Solving related rates problems can feel daunting, but a systematic approach can make the process more manageable. Here's a breakdown of the key steps:

1. Read the Problem Carefully: Understand what the problem is asking. Identify the given information and what you need to find. Highlight keywords and quantities.
2. Draw a Diagram: If the problem involves a geometric shape or a physical scenario, draw a diagram. This visual representation can help you understand the relationships between the variables. Label all relevant quantities.
3. Identify Variables and Rates: Assign variables to the quantities that are changing. Identify the rates of change that are given (e.g., dx/dt, dy/dt) and the rate you are trying to find.
4. Establish a Relationship: Find an equation that relates the variables. This equation might come from a geometric formula (e.g., area of a circle, volume of a cone), the Pythagorean theorem, or some other relationship described in the problem.
5. Differentiate Implicitly: Differentiate both sides of the equation with respect to time (t). Remember to use the chain rule when differentiating terms involving variables other than t.
6. Substitute Known Values: Substitute the given values for the variables and rates into the differentiated equation.
7. Solve for the Unknown Rate: Solve the equation for the rate of change that you are trying to find.
8. State the Answer with Units: Make sure to include the correct units in your answer. For example, if you are finding a rate of change of distance, the units might be meters per second (m/s) or miles per hour (mph).

Example Problems and Solutions

Let's illustrate these steps with some example problems similar to those you might encounter in Pearson Calculus CP section 3.6.

Example 1: Expanding Circle

A circular puddle is expanding at a rate of 3 cm/s. At what rate is the area of the puddle increasing when the radius is 10 cm?

1. Read the Problem: We are given the rate of change of the radius (dr/dt) and asked to find the rate of change of the area (dA/dt) when r = 10 cm.

2. Draw a Diagram: Draw a circle representing the puddle. Label the radius as r and the area as A.

3. Identify Variables and Rates:

   • r = radius of the puddle (cm)
   • A = area of the puddle (cm²)
   • dr/dt = 3 cm/s (rate of change of radius)
   • dA/dt = ? (rate of change of area – what we want to find)

4. Establish a Relationship: The area of a circle is given by A = πr².

5. Differentiate Implicitly: Differentiate both sides of the equation with respect to t:

   • dA/dt = 2πr (dr/dt)

6. Substitute Known Values: Substitute r = 10 cm and dr/dt = 3 cm/s:

   • dA/dt = 2π(10)(3)

7. Solve for the Unknown Rate:

   • dA/dt = 60π

8. State the Answer with Units: The area of the puddle is increasing at a rate of 60π cm²/s.

Example 2: Sliding Ladder

A 10-foot ladder leans against a wall. If the bottom of the ladder slides away from the wall at a rate of 2 ft/s, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 feet from the wall?

1. Read the Problem: We are given the length of the ladder, the rate at which the bottom is sliding away (dx/dt), and we need to find the rate at which the top is sliding down (dy/dt) when the bottom is 6 feet from the wall.

2. Draw a Diagram: Draw a right triangle with the ladder as the hypotenuse, the wall as one leg (y), and the distance from the wall to the bottom of the ladder as the other leg (x).

3. Identify Variables and Rates:

   • x = distance from the wall to the bottom of the ladder (ft)
   • y = distance from the top of the ladder to the ground (ft)
   • dx/dt = 2 ft/s (rate at which the bottom is sliding away)
   • dy/dt = ? (rate at which the top is sliding down – what we want to find)
   • Ladder length = 10 ft (constant)

4. Establish a Relationship: The Pythagorean theorem applies: x² + y² = 10².

5. Differentiate Implicitly: Differentiate both sides of the equation with respect to t:

   • 2x (dx/dt) + 2y (dy/dt) = 0

6. Substitute Known Values: We are given x = 6 ft and dx/dt = 2 ft/s. We need to find y when x = 6. Using the Pythagorean theorem:

   • 6² + y² = 10²
   • y² = 100 - 36 = 64
   • y = 8 ft

   Now substitute the values into the differentiated equation:

   • 2(6)(2) + 2(8)(dy/dt) = 0

7. Solve for the Unknown Rate:

   • 24 + 16 (dy/dt) = 0
   • 16 (dy/dt) = -24
   • dy/dt = -24/16 = -3/2

8. State the Answer with Units: The top of the ladder is sliding down the wall at a rate of 3/2 ft/s (or 1.5 ft/s). The negative sign indicates that y is decreasing.

Example 3: Filling a Conical Tank

Water is flowing into a conical tank at a rate of 2 m³/min. The tank is 4 m tall and has a radius of 2 m at the top. How fast is the water level rising when the water is 3 m deep?

1. Read the Problem: We are given the rate at which the volume is changing (dV/dt), the dimensions of the cone, and we need to find the rate at which the water level is rising (dh/dt) when the water is 3 m deep.

2. Draw a Diagram: Draw a cone representing the tank. Label the height as h, the radius of the water surface as r, and the volume of the water as V.

3. Identify Variables and Rates:

   • V = volume of the water in the tank (m³)
   • h = height of the water in the tank (m)
   • r = radius of the water surface (m)
   • dV/dt = 2 m³/min (rate at which water is flowing in)
   • dh/dt = ? (rate at which the water level is rising – what we want to find)
   • Tank height = 4 m (constant)
   • Tank radius = 2 m (constant)

4. Establish a Relationship: The volume of a cone is given by V = (1/3)πr²h. However, we have two variables, r and h, both of which are changing. We need to relate r and h using similar triangles. The ratio of the radius to the height of the tank is constant: r/h = 2/4 = 1/2. Therefore, r = h/2.

   Substitute this into the volume equation:

   • V = (1/3)π(h/2)²h = (1/3)π(h²/4)h = (1/12)πh³

5. Differentiate Implicitly: Differentiate both sides of the equation with respect to t:

   • dV/dt = (1/4)πh² (dh/dt)

6. Substitute Known Values: We are given dV/dt = 2 m³/min and h = 3 m:

   • 2 = (1/4)π(3)² (dh/dt)

7. Solve for the Unknown Rate:

   • 2 = (9/4)π (dh/dt)
   • dh/dt = 8/(9π)

8. State the Answer with Units: The water level is rising at a rate of 8/(9π) m/min.

Common Challenges and How to Overcome Them

• Identifying the Correct Equation: The biggest challenge is often finding the correct equation that relates the variables. Carefully analyze the problem and draw a diagram. Consider geometric formulas, trigonometric relationships, and other relevant equations.
• Implicit Differentiation: Remember to use the chain rule when differentiating terms involving variables other than t. Treat each variable as a function of t.
• Units: Always include units in your answer. Double-check that your units are consistent throughout the problem.
• Algebraic Manipulation: Solving for the unknown rate often involves algebraic manipulation. Practice your algebra skills to avoid errors.
• Understanding the Problem: If you're struggling to understand the problem, try rephrasing it in your own words. Draw a diagram and label all relevant quantities.

Strategies for Success

• Practice Regularly: The best way to master related rates problems is to practice them regularly. Work through as many problems as possible, and don't be afraid to ask for help when you get stuck.
• Work Through Examples: Study worked examples carefully. Pay attention to the steps involved and the reasoning behind each step.
• Draw Diagrams: Visualizing the problem with a diagram can make it easier to understand the relationships between the variables.
• Check Your Answers: If possible, check your answers by plugging them back into the original equation. Does your answer make sense in the context of the problem?
• Collaborate with Others: Work with classmates or form a study group. Explaining concepts to others can help you solidify your understanding.
• Seek Help When Needed: Don't be afraid to ask your teacher or a tutor for help if you're struggling. Early intervention can prevent you from falling behind.

Advanced Techniques and Considerations

While the basic steps outlined above are sufficient for most problems in Pearson Calculus CP section 3.6, some problems may require more advanced techniques.

• Optimization Problems: Some related rates problems may involve finding the maximum or minimum value of a quantity. These problems often require using calculus to find critical points and then applying the first or second derivative test to determine whether they are maxima or minima.
• Trigonometric Functions: Some problems may involve trigonometric functions. Remember the derivatives of trigonometric functions and how to apply the chain rule when differentiating them.
• Multiple Related Rates: Some problems may involve multiple related rates. In these cases, you may need to set up a system of equations and solve for the unknown rates.

FAQs About Pearson 3.6 Exercises

• Q: Why are related rates problems so difficult?

  • A: Related rates problems require you to combine several calculus concepts, including implicit differentiation, the chain rule, and geometric formulas. They also require strong problem-solving skills and the ability to translate word problems into mathematical equations.

• Q: What is the most common mistake students make on related rates problems?

  • A: One of the most common mistakes is failing to differentiate implicitly correctly. Remember to use the chain rule when differentiating terms involving variables other than t.

• Q: How can I improve my problem-solving skills for related rates problems?

  • A: The best way to improve your problem-solving skills is to practice regularly. Work through as many problems as possible, and don't be afraid to ask for help when you get stuck.

• Q: Are there any online resources that can help me with related rates problems?

  • A: Yes, there are many online resources available, including websites, videos, and practice problems. Khan Academy and Paul's Online Math Notes are excellent starting points. Additionally, search for videos on YouTube specifically addressing related rates problems.

• Q: What if I am stuck on a specific problem?

  • A: First, reread the problem carefully and make sure you understand what it is asking. Draw a diagram and label all relevant quantities. Try to identify the equation that relates the variables. If you are still stuck, ask your teacher or a tutor for help.

Conclusion

Mastering the exercises in Pearson Calculus CP section 3.6 requires a solid understanding of related rates concepts and a systematic problem-solving approach. By carefully reading the problem, drawing a diagram, identifying variables and rates, establishing a relationship, differentiating implicitly, substituting known values, and solving for the unknown rate, you can tackle even the most challenging problems. Remember to practice regularly, work through examples, and seek help when needed. With consistent effort, you can conquer related rates and excel in Calculus CP. The journey through calculus, while challenging, is rewarding, opening doors to deeper understanding and application in various fields. Embrace the challenge, and you'll find yourself mastering not just calculus, but also the art of problem-solving itself.