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Practice Questions
Test your understanding with targeted questions related to the topic.
Question 1
Easy
Integrate \( \frac{\partial z}{\partial x} = 5x \).
💡 Hint: Remember to treat y as a constant.
Question 2
Easy
What is the result of integrating \( \frac{\partial z}{\partial y} = 3y^2 \)?
💡 Hint: Think about what happens to x during this integration.
Practice 4 more questions and get performance evaluation
Interactive Quizzes
Engage in quick quizzes to reinforce what you've learned and check your comprehension.
Question 1
What is direct integration used for?
- Solving ordinary differential equations
- Solving Partial Differential Equations
- Finding limits of functions
💡 Hint: Think about which type of equations involve partial derivatives.
Question 2
When given \( \frac{\partial z}{\partial x} = f(x,y) \), we integrate with respect to which variable?
- True
- False
💡 Hint: This ensures we correctly find \\( z \\) as a function of both variables.
Solve and get performance evaluation
Challenge Problems
Push your limits with challenges.
Question 1
Given the PDEs \( \frac{\partial z}{\partial x} = 3x^2 \) and \( \frac{\partial z}{\partial y} = 4y \), derive the function z.
💡 Hint: Remember to add both arbitrary functions after integration.
Question 2
For the PDE \( \frac{\partial z}{\partial x} = sin(xy) \), integrate to find z where y is treated as constant.
💡 Hint: Recall how we adjust our integration based on treating y as a constant.
Challenge and get performance evaluation