Computing ∂f/∂x (0,0) | Midterm Review Sheet Problem 2(a)

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In summary, the conversation is discussing problem 2(a) from a midterm review sheet for a math class. The student is unsure about what equations are needed to solve the problem and is considering using the quotient rule. They mention that they cannot simply plug in (0,0) and are wondering if there is a theorem they can use. They also mention that they think they have found a solution by evaluating ∂f/∂x|x=0 and then (∂f/∂x|x=0)|y=0.
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Jamin2112
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Homework Statement



This is problem 2(a) from my midterm review sheet, found here: http://www.math.washington.edu/~sullivan/326smt_sp10.pdf

Homework Equations



Not sure what exactly I need here

The Attempt at a Solution



By the quotient rule, ∂f/∂x = [(x2+y2)(y2)-(xy2)(2x)]/[x2+y2]2. Obviously I can't just plug in (0,0). I thought about computing lim(x,y)-->(0,0) f(x,y), but the limit does exist, and doing that seems to be saved for part (b). Is there some theorem I should use to compute ∂f/∂x|(0,0)?

Thank you for for help.
 
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OH! I think I might be on to something. I could first evaluate ∂f/∂x|x=0 , and then (∂f/∂x|x=0)|y=0.

∂f/∂x|x=0 = y4/y4 = 1 ---> (∂f/∂x|x=0)|y=0 = 1.

Right?
 

1. What does "Computing ∂f/∂x (0,0) | Midterm Review Sheet Problem 2(a)" mean?

Computing ∂f/∂x (0,0) | Midterm Review Sheet Problem 2(a) is a mathematical notation that represents finding the partial derivative of a function f with respect to the variable x at the point (0,0). It is a common problem in calculus and is often used in the study of multivariable functions.

2. How do you compute ∂f/∂x (0,0)?

To compute ∂f/∂x (0,0), you will need to use the limit definition of the partial derivative. This involves taking the limit of a difference quotient as the change in x approaches 0. You will also need to use the chain rule and product rule if the function is a composition or a product of multiple functions. It is important to carefully follow the steps and simplify as much as possible to find the final answer.

3. What is the significance of finding ∂f/∂x (0,0)?

Finding ∂f/∂x (0,0) allows you to determine the rate of change of a multivariable function f with respect to the variable x at a specific point (0,0). This can be useful in understanding the behavior of the function and making predictions about its values at nearby points. It is also an important concept in optimization problems, where finding the critical points (points where the partial derivatives are equal to 0) can help identify the maximum or minimum values of a function.

4. What are some common mistakes when computing ∂f/∂x (0,0)?

Some common mistakes when computing ∂f/∂x (0,0) include forgetting to use the limit definition, not simplifying the difference quotient, and making errors in applying the chain rule or product rule. It is also important to carefully plug in the values of x and y at the point (0,0) and to pay attention to the order of operations. It is recommended to double-check all steps and simplify as much as possible to avoid mistakes.

5. Can you provide an example of computing ∂f/∂x (0,0)?

Yes, for example, let f(x,y) = xy2 + x2y. To find ∂f/∂x (0,0), we will use the limit definition: ∂f/∂x (0,0) = limh→0 (f(0+h,0) - f(0,0))/h = limh→0 ((0+h)02 + (0+h)20 - 0)/h = limh→0 (0 + 0 - 0)/h = 0. Therefore, ∂f/∂x (0,0) = 0.

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