Directional Derivatives Quiz

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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)?

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?

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)?

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)?

x = 1 - 2t, y = -t, z = 2 - t Signup and view all the answers

What is the gradient of f at the point P(2, -1) where f(x, y) = x³y + y³x - 2xy?

(-11, -16) Signup and view all the answers

What is the equation of the tangent plane to the surface f(x, y) = x / (x + y) at the point (x₀, y₀) = (1, 1)?

x - y - 4z + 2 = 0 Signup and view all the answers

What is the first derivative of z with respect to t in terms of f, φ, and u?

f'xφ' + f'y*u' Signup and view all the answers

What is the second derivative of z with respect to t in terms of f, φ, and u?

f''x(φ')^2 + 2f''xyphi'u' + f''y(u')^2 + f'xφ'' + f'yu'' Signup and view all the answers

What is the directional derivative of f(x, y) = xy at (x0, y0) in the direction of (h, k) = (1, 1)?

√2 Signup and view all the answers

What is the directional derivative of f(x, y) = 2x + y - 1 at (2, 1) in the direction of (1, 1)?

3√2/2 Signup and view all the answers

What is the directional derivative of g(x, y) = xe^(yx) - xy at (1, 1) in the direction of (3, 4)?

2e - 7/5 Signup and view all the answers

What is the directional derivative of f(x, y) = x + 2x^2 - 3xy at (1, 1) in the direction of (3/5, 4/5)?

-6/5 Signup and view all the answers

What is the partial derivative of 𝑥1 with respect to 𝑝1?

−0.4𝑝1−1.4 𝑒0.2𝑝2 Signup and view all the answers

What is the relationship between 𝑋1 and 𝑋2 based on the given demand functions?

They are substitute goods. Signup and view all the answers

What is the partial derivative of 𝑥2 with respect to 𝑝2?

−0.6𝑝2−1.6 𝑒0.5𝑝1 Signup and view all the answers

What is the formula for the partial price elasticity of demand?

−𝑎11 𝑝2 + 𝑎1 𝑒 Signup and view all the answers

What is the demand function for 𝑥1 in the second scenario?

𝑥1 = 2𝑝1−0.6 𝑝20.8 Signup and view all the answers

What type of goods are 𝑋1 and 𝑋2 in the second scenario?

Substitute goods Signup and view all the answers

What is the condition for a function 𝑓(𝑥, 𝑦) to be homogeneous of degree 𝑟?

𝑓(𝑥, 𝑦) = 𝑥 𝑟 𝜓(𝑦/𝑥) Signup and view all the answers

What is the value of cross-price elasticity of good 1 with respect to the price of good 2?

0.8 Signup and view all the answers

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?

1.6% Signup and view all the answers

What is the degree of homogeneity of the function 𝑧 = 𝑥 2 + 2𝑥 2 𝑦 2 + 𝑦 4 /𝑥 + 𝑦?

2 Signup and view all the answers

What is the value of own-price elasticity of good 2?

-0.5 Signup and view all the answers

What is the property of the first partial derivatives of a homogeneous function of degree 𝑟?

They are homogeneous of degree 𝑟 - 1 Signup and view all the answers

What type of goods are good 1 and good 2 based on the given demand functions?

Competitive Signup and view all the answers

What is the Euler's theorem for a homogeneous function of degree 𝑟?

𝑥(𝜕𝑧/𝜕𝑥) + 𝑦(𝜕𝑧/𝜕𝑦) = 𝑟𝑧 Signup and view all the answers

Determine the degree of homogeneity of the function 𝑧 = 2(𝑎𝑥 + 𝑏𝑦)2 - (𝑥 2 + 𝑦 2 )?

2 Signup and view all the answers

What is the value of cross-price elasticity of good 2 with respect to the price of good 1?

0.7 Signup and view all the answers

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?

1.4% Signup and view all the answers

What is the condition for a function 𝑓(𝑥, 𝑦) to be homogeneous of degree 0?

𝑓(𝑥, 𝑦) = 𝜓(𝑦/𝑥) Signup and view all the answers