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What is the equation of the tangent plane at the point (1, 0, 0) to the surface z = ln(x² + y²)?
What is the equation of the tangent plane at the point (1, 0, 0) to the surface z = ln(x² + y²)?
- z = x - y - 1
- z = x + y - 1
- z = 2x + y - 1
- z = x - 1 (correct)
What is the equation of the tangent plane at the point (1, 1, 5) to the graph of f(x, y) = x² + 2xy + 2y²?
What is the equation of the tangent plane at the point (1, 1, 5) to the graph of f(x, y) = x² + 2xy + 2y²?
- z = 5x + 3y - 2
- z = 3x + 2y - 4
- z = 2x + 4y - 3
- z = 4x + 6y - 5 (correct)
What is the equation of the tangent plane at the point (x₁, y₁, z₁) to the surface z = √(xy)?
What is the equation of the tangent plane at the point (x₁, y₁, z₁) to the surface z = √(xy)?
- y₁x + x₁y - 2z₁z = 0 (correct)
- y₁x - x₁y + z₁z = 0
- 2y₁x + x₁y - z₁z = 0
- x₁y + y₁x - z₁z = 0
What is the equation of the normal line to the surface f(x, y) = x² + y⁴ + e^(xy) at the point (1, 0, 2)?
What is the equation of the normal line to the surface f(x, y) = x² + y⁴ + e^(xy) at the point (1, 0, 2)?
What is the gradient of f at the point P(2, -1) where f(x, y) = x³y + y³x - 2xy?
What is the gradient of f at the point P(2, -1) where f(x, y) = x³y + y³x - 2xy?
What is the equation of the tangent plane to the surface f(x, y) = x / (x + y) at the point (x₀, y₀) = (1, 1)?
What is the equation of the tangent plane to the surface f(x, y) = x / (x + y) at the point (x₀, y₀) = (1, 1)?
What is the first derivative of z with respect to t in terms of f, φ, and u?
What is the first derivative of z with respect to t in terms of f, φ, and u?
What is the second derivative of z with respect to t in terms of f, φ, and u?
What is the second derivative of z with respect to t in terms of f, φ, and u?
What is the directional derivative of f(x, y) = xy at (x0, y0) in the direction of (h, k) = (1, 1)?
What is the directional derivative of f(x, y) = xy at (x0, y0) in the direction of (h, k) = (1, 1)?
What is the directional derivative of f(x, y) = 2x + y - 1 at (2, 1) in the direction of (1, 1)?
What is the directional derivative of f(x, y) = 2x + y - 1 at (2, 1) in the direction of (1, 1)?
What is the directional derivative of g(x, y) = xe^(yx) - xy at (1, 1) in the direction of (3, 4)?
What is the directional derivative of g(x, y) = xe^(yx) - xy at (1, 1) in the direction of (3, 4)?
What is the directional derivative of f(x, y) = x + 2x^2 - 3xy at (1, 1) in the direction of (3/5, 4/5)?
What is the directional derivative of f(x, y) = x + 2x^2 - 3xy at (1, 1) in the direction of (3/5, 4/5)?
What is the partial derivative of 𝑥1 with respect to 𝑝1?
What is the partial derivative of 𝑥1 with respect to 𝑝1?
What is the relationship between 𝑋1 and 𝑋2 based on the given demand functions?
What is the relationship between 𝑋1 and 𝑋2 based on the given demand functions?
What is the partial derivative of 𝑥2 with respect to 𝑝2?
What is the partial derivative of 𝑥2 with respect to 𝑝2?
What is the formula for the partial price elasticity of demand?
What is the formula for the partial price elasticity of demand?
What is the demand function for 𝑥1 in the second scenario?
What is the demand function for 𝑥1 in the second scenario?
What type of goods are 𝑋1 and 𝑋2 in the second scenario?
What type of goods are 𝑋1 and 𝑋2 in the second scenario?
What is the condition for a function 𝑓(𝑥, 𝑦) to be homogeneous of degree 𝑟?
What is the condition for a function 𝑓(𝑥, 𝑦) to be homogeneous of degree 𝑟?
What is the value of cross-price elasticity of good 1 with respect to the price of good 2?
What is the value of cross-price elasticity of good 1 with respect to the price of good 2?
What is the percentage change in the quantity demanded of good 1 when the price of good 1 increases by 2% from the existing level?
What is the percentage change in the quantity demanded of good 1 when the price of good 1 increases by 2% from the existing level?
What is the degree of homogeneity of the function 𝑧 = 𝑥 2 + 2𝑥 2 𝑦 2 + 𝑦 4 /𝑥 + 𝑦?
What is the degree of homogeneity of the function 𝑧 = 𝑥 2 + 2𝑥 2 𝑦 2 + 𝑦 4 /𝑥 + 𝑦?
What is the value of own-price elasticity of good 2?
What is the value of own-price elasticity of good 2?
What is the property of the first partial derivatives of a homogeneous function of degree 𝑟?
What is the property of the first partial derivatives of a homogeneous function of degree 𝑟?
What type of goods are good 1 and good 2 based on the given demand functions?
What type of goods are good 1 and good 2 based on the given demand functions?
What is the Euler's theorem for a homogeneous function of degree 𝑟?
What is the Euler's theorem for a homogeneous function of degree 𝑟?
Determine the degree of homogeneity of the function 𝑧 = 2(𝑎𝑥 + 𝑏𝑦)2 - (𝑥 2 + 𝑦 2 )?
Determine the degree of homogeneity of the function 𝑧 = 2(𝑎𝑥 + 𝑏𝑦)2 - (𝑥 2 + 𝑦 2 )?
What is the value of cross-price elasticity of good 2 with respect to the price of good 1?
What is the value of cross-price elasticity of good 2 with respect to the price of good 1?
What is the percentage change in the quantity demanded of good 2 when the price of good 2 decreases by 2% from the existing level?
What is the percentage change in the quantity demanded of good 2 when the price of good 2 decreases by 2% from the existing level?
What is the condition for a function 𝑓(𝑥, 𝑦) to be homogeneous of degree 0?
What is the condition for a function 𝑓(𝑥, 𝑦) to be homogeneous of degree 0?
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