Math Problem Statement
Solution
Let's analyze the provided image and solve the problems.
Problem 8:
Given the function defined as:
\frac{7x^2 + y^2}{x^2 + y^2} & \text{if } (x, y) \neq (0,0), \\ 0 & \text{if } (x, y) = (0,0), \end{cases}$$ Options: a) $$f(x, 0)$$ does not exist. b) $$f(0, y) = 7$$. c) $$f(0, 0)$$ does not exist. d) $$f(0, 0) = 0$$. To check the correctness, let's analyze the function. 1. For $$(x, y) \neq (0,0)$$: $$f(x, y) = \frac{7x^2 + y^2}{x^2 + y^2}$$ For specific values: - If $$y = 0$$: $$f(x, 0) = \frac{7x^2}{x^2} = 7$$ - If $$x = 0$$: $$f(0, y) = \frac{y^2}{y^2} = 1$$ 2. At $$(x, y) = (0,0)$$: The function is defined as: $$f(0,0) = 0$$ Now let's examine each option: - **Option a)** $$f(x, 0)$$ does not exist. - This is incorrect, $$f(x, 0) = 7$$ for all $$x \neq 0$$. - **Option b)** $$f(0, y) = 7$$. - This is incorrect, $$f(0, y) = 1$$ for all $$y \neq 0$$. - **Option c)** $$f(0, 0)$$ does not exist. - This is incorrect, $$f(0, 0) = 0$$. - **Option d)** $$f(0, 0) = 0$$. - This is correct. Therefore, the correct option for problem 8 is **d)**. ### Problem 9: Given the function $$f(x, y)$$ defined as: $$f(x, y) = \begin{cases} \frac{x^2 y - x^3 y^2}{x^2 + y^2} & \text{if } (x, y) \neq (0,0), \\ 0 & \text{if } (x, y) = (0,0), \end{cases}$$ Options: a) $$f(0,0) = 1$$ and $$f(0,0) = 2$$. b) $$f(0,0) = 0$$ and $$f(x,y) = 0$$. c) $$f(0,0) = 0$$ and $$f(x,0) = 0$$. d) $$f(0,0) = 0$$ and $$f(0,y) = 2$$. To check the correctness, let's analyze the function. 1. For $$(x, y) \neq (0,0)$$: $$f(x, y) = \frac{x^2 y - x^3 y^2}{x^2 + y^2}$$ For specific values: - If $$y = 0$$: $$f(x, 0) = \frac{x^2 \cdot 0 - x^3 \cdot 0^2}{x^2 + 0^2} = 0$$ - If $$x = 0$$: $$f(0, y) = \frac{0 \cdot y - 0 \cdot y^2}{0 + y^2} = 0$$ 2. At $$(x, y) = (0,0)$$: The function is defined as: $$f(0,0) = 0$$ Now let's examine each option: - **Option a)** $$f(0,0) = 1$$ and $$f(0,0) = 2$$. - This is incorrect, $$f(0,0) = 0$$. - **Option b)** $$f(0,0) = 0$$ and $$f(x,y) = 0$$. - This is incorrect, $$f(x,y) \neq 0$$ for all $$(x,y) \neq (0,0)$$. - **Option c)** $$f(0,0) = 0$$ and $$f(x,0) = 0$$. - This is correct, since we found $$f(x,0) = 0$$. - **Option d)** $$f(0,0) = 0$$ and $$f(0,y) = 2$$. - This is incorrect, $$f(0,y) = 0$$ for all $$y \neq 0$$. Therefore, the correct option for problem 9 is **c)**. ### Summary: - Problem 8: Correct answer is **d)**. - Problem 9: Correct answer is **c)**. Would you like more details on these solutions or have any other questions? Here are some related questions: 1. What is the limit definition of a function's continuity? 2. How can we evaluate the limit of a function with two variables? 3. What is the difference between pointwise and uniform convergence? 4. How can you check if a multivariable function is continuous at a point? 5. What are the different types of discontinuities? 6. How do we differentiate a function of several variables? 7. What are partial derivatives and how are they used? 8. How do we find the gradient of a function with several variables? **Tip**: Always check the definition of a piecewise function at the boundary points to ensure continuity.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Limits
Continuity
Formulas
Piecewise function definition
Limit definition
Theorems
Definition of continuity
Suitable Grade Level
High School