CAN SOMEONE PLEASE TELL ME HOW THESE TYPE OF QUESTIONS ARE SOLVED. A STEP BY STEP GUIDE PLEASEEEE Figure 1 Shows Part Of The Curve With Equation y = X/2 + 4/xsquare in The Interval 0.8 < X < 7 By Drawing A Suitable Straight Line On The Grid, Obtain

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CAN SOMEONE PLEASE TELL ME HOW THESE TYPE OF QUESTIONS ARE SOLVED. A STEP BY STEP GUIDE PLEASEEEE

The problem presents a curve with a given equation, y = x/2 + 4/x^2, in the interval 0.8 < x < 7. The task is to draw a suitable straight line on the grid to represent the curve in the given interval. This problem requires a step-by-step approach to understand the curve's behavior and to draw an accurate representation of it.

Step 1: Analyze the Equation

The given equation is y = x/2 + 4/x^2. To understand the curve's behavior, we need to analyze the equation and identify its key features.

  • Domain: The domain of the function is the set of all possible input values (x) for which the function is defined. In this case, the domain is 0.8 < x < 7.
  • Range: The range of the function is the set of all possible output values (y) for which the function is defined. To determine the range, we need to find the minimum and maximum values of the function in the given interval.
  • Critical Points: Critical points are the values of x that make the derivative of the function equal to zero or undefined. These points can help us identify the curve's behavior and determine the intervals where the curve is increasing or decreasing.

Step 2: Find the Derivative

To find the derivative of the function, we'll apply the power rule and the quotient rule.

y = x/2 + 4/x^2

Using the quotient rule, we get:

y' = (1/2 - 8/x^3) / 1

Simplifying the derivative, we get:

y' = (x^3 - 16) / 2x^3

Step 3: Find the Critical Points

To find the critical points, we'll set the derivative equal to zero and solve for x.

(x^3 - 16) / 2x^3 = 0

Multiplying both sides by 2x^3, we get:

x^3 - 16 = 0

Adding 16 to both sides, we get:

x^3 = 16

Taking the cube root of both sides, we get:

x = 2.5198421 (approximately)

Since x = 2.5198421 is not in the given interval 0.8 < x < 7, we'll ignore this critical point.

Step 4: Determine the Intervals of Increase and Decrease

To determine the intervals of increase and decrease, we'll use the first derivative test.

  • Interval 1: 0.8 < x < 2.5198421
    • The derivative is positive in this interval, so the function is increasing.
  • Interval 2: 2.5198421 < x < 7
    • The derivative is negative in this interval, so the function is decreasing.

Step 5: Draw the Curve

Using the information gathered from the previous steps, we can draw the curve on the grid.

  • Interval 1: 0.8 < x < 2.5198421
    • The curve is increasing in this interval.
  • Interval 2: 2.5198421 < x < 7
    • The curve is decreasing in this interval.

Conclusion

In this step-by-step guide, we analyzed the given equation, found the derivative, determined the critical points, and determined the intervals of increase and decrease. Using this information, we drew the curve on the grid, representing the curve in the given interval.

Key Takeaways

  • To solve this type of problem, we need to analyze the equation, find the derivative, determine the critical points, and determine the intervals of increase and decrease.
  • The first derivative test can help us determine the intervals of increase and decrease.
  • Drawing the curve on the grid requires a good understanding of the curve's behavior in the given interval.

Final Thoughts

Solving this type of problem requires a step-by-step approach, analyzing the equation, finding the derivative, determining the critical points, and determining the intervals of increase and decrease. By following these steps, we can draw an accurate representation of the curve on the grid.
Frequently Asked Questions (FAQs) - Solving Curve Representation Problems

Q: What is the first step in solving a curve representation problem?

A: The first step in solving a curve representation problem is to analyze the given equation and understand its key features, such as the domain, range, and critical points.

Q: How do I find the derivative of a function?

A: To find the derivative of a function, you can use the power rule and the quotient rule. The power rule states that if y = x^n, then y' = nx^(n-1). The quotient rule states that if y = u/v, then y' = (vu' - uv')/v^2.

Q: What are critical points, and how do I find them?

A: Critical points are the values of x that make the derivative of the function equal to zero or undefined. To find critical points, you can set the derivative equal to zero and solve for x.

Q: How do I determine the intervals of increase and decrease?

A: To determine the intervals of increase and decrease, you can use the first derivative test. If the derivative is positive, the function is increasing. If the derivative is negative, the function is decreasing.

Q: What is the significance of the first derivative test?

A: The first derivative test is a powerful tool for determining the intervals of increase and decrease. It helps us understand the behavior of the function and make informed decisions about drawing the curve on the grid.

Q: How do I draw the curve on the grid?

A: To draw the curve on the grid, you need to use the information gathered from the previous steps. You can use the intervals of increase and decrease to determine the shape of the curve and the critical points to determine the location of the curve.

Q: What are some common mistakes to avoid when solving curve representation problems?

A: Some common mistakes to avoid when solving curve representation problems include:

  • Not analyzing the equation thoroughly
  • Not finding the derivative correctly
  • Not determining the critical points correctly
  • Not using the first derivative test correctly
  • Not drawing the curve accurately on the grid

Q: How can I practice solving curve representation problems?

A: You can practice solving curve representation problems by:

  • Working through example problems
  • Using online resources and tutorials
  • Practicing with different types of functions and equations
  • Joining a study group or seeking help from a tutor

Q: What are some real-world applications of curve representation problems?

A: Curve representation problems have many real-world applications, including:

  • Physics and engineering: Curve representation is used to model the motion of objects and predict their behavior.
  • Economics: Curve representation is used to model economic systems and predict economic trends.
  • Computer science: Curve representation is used in computer graphics and game development to create realistic models and animations.

Conclusion

Solving curve representation problems requires a step-by-step approach, analyzing the equation, finding the derivative, determining the critical points, and determining the intervals of increase and decrease. By following these steps and avoiding common mistakes, you can draw an accurate representation of the curve on the grid. Practice and real-world applications can help you become proficient in solving curve representation problems.