Calculate Ascending Area: Finding Area Under f(x)=2x²

Q: What does 'ascending area' actually mean?

The term 'ascending area' refers to the intervals where the function is increasing (going upward). It's not about calculating actual area under the curve - it's about finding where the function rises!

Q: Why do I need to find the derivative first?

The derivative tells you the slope at any point. When $ f'(x) > 0 $, the slope is positive, meaning the function is increasing at that point.

Q: How do I solve 4x > 0?

Divide both sides by 4: $ x > 0 $. Since 4 is positive, the inequality direction stays the same. This means the function increases for all positive x values.

Q: What happens at x = 0?

At $ x = 0 $, we have $ f'(0) = 4(0) = 0 $. The derivative is zero, so the function is neither increasing nor decreasing - it's at a turning point!

Q: Can I just look at the graph instead?

Yes! For $ f(x) = 2x^2 $, the graph is a U-shaped parabola. You can see it decreases for $ x and increases for \( x > 0 $, but using derivatives is more precise.

Function Derivatives with Increasing Intervals

Find the ascending area of the function

$f(x)=2x^2$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the domain of increase of the function

00:04 We'll examine the coefficient of X squared, it's positive so the function is happy

00:08 We'll examine the trinomial coefficients

00:16 We'll use the formula to find the vertex of the parabola

00:21 We'll substitute appropriate values according to the given data and solve to find the vertex

00:27 This is the X value at the vertex of the parabola

00:32 We'll determine based on the type of parabola when it's decreasing and when it's increasing

00:37 We'll draw the X-axis and find the domain of increase

00:41 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Find the ascending area of the function

$f(x)=2x^2$

Step-by-step solution

To determine the intervals where the function $f(x) = 2x^2$ is increasing, we will analyze the derivative of the function:

Step 1: Differentiate the function.
The derivative of $f(x) = 2x^2$ is $f'(x) = 4x$ .

Step 2: Determine where $f'(x) > 0$ .
To find the increasing intervals, set $4x > 0$ . Solving this inequality, we obtain $x > 0$ .

Therefore, the function $f(x) = 2x^2$ is increasing for $x > 0$ .

Consequently, the correct answer is the interval where the function is increasing, which is $0 < x$ .

Final Answer

$0 < x$

Key Points to Remember

Essential concepts to master this topic

Rule: Find where f'(x) > 0 to determine increasing intervals
Technique: Derivative of $2x^2$ is $4x$ , solve $4x > 0$
Check: Test values: at x = 1, f'(1) = 4 > 0, so increasing ✓

Common Mistakes

Avoid these frequent errors

Confusing increasing with area under curve
Don't calculate integrals or areas when asked for increasing intervals = wrong concept entirely! The question asks where the function is ascending (going up), not area. Always find where the derivative is positive to identify increasing regions.

Practice Quiz

Test your knowledge with interactive questions

Find the ascending area of the function

$$ f(x)=2x^2 $$

FAQ

Everything you need to know about this question

What does 'ascending area' actually mean?

The term 'ascending area' refers to the intervals where the function is increasing (going upward). It's not about calculating actual area under the curve - it's about finding where the function rises!

Why do I need to find the derivative first?

The derivative tells you the slope at any point. When $f'(x) > 0$ , the slope is positive, meaning the function is increasing at that point.

How do I solve 4x > 0?

Divide both sides by 4: $x > 0$ . Since 4 is positive, the inequality direction stays the same. This means the function increases for all positive x values.

What happens at x = 0?

At $x = 0$ , we have $f'(0) = 4(0) = 0$ . The derivative is zero, so the function is neither increasing nor decreasing - it's at a turning point!

Can I just look at the graph instead?

Yes! For $f(x) = 2x^2$ , the graph is a U-shaped parabola. You can see it decreases for $x < 0$ and increases for $x > 0$ , but using derivatives is more precise.

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Calculate Ascending Area: Finding Area Under f(x)=2x²

Function Derivatives with Increasing Intervals

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

What does 'ascending area' actually mean?

Why do I need to find the derivative first?

How do I solve 4x > 0?

What happens at x = 0?

Can I just look at the graph instead?

🌟 Unlock Your Math Potential

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