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Questions and Answers
Which equation represents the surface modeled by the equation $y^2 + 4x^2 - 2z^2 = 1$?
Which equation represents the surface modeled by the equation $y^2 + 4x^2 - 2z^2 = 1$?
- $z = 1 - 4x^2 - 2y^2$
- $z = 1 + 4x^2 + 2y^2$
- $z = 1 - 2x^2 - y^2$ (correct)
- $z = 1 + 2x^2 + y^2$
What is the value of $y^3 + x^3$ when evaluated along the path $y = mx$, where $m > 0$?
What is the value of $y^3 + x^3$ when evaluated along the path $y = mx$, where $m > 0$?
- $2m^3$
- $3m^3$
- $4m^3$
- $m^3$ (correct)
What is the linear approximation of $f(0.8, 0.8)$ for the function $f(x, y) = 4\cos(2x - y)$ at $(\pi/4, \pi/4)$?
What is the linear approximation of $f(0.8, 0.8)$ for the function $f(x, y) = 4\cos(2x - y)$ at $(\pi/4, \pi/4)$?
- $4\sqrt{2}$
- $2\sqrt{2}$ (correct)
- $4$
- $2$
What are the absolute extrema of $f(x, y) = x^2y - y^3$ on the triangle bounded by $0 \leq x \leq 2$ and $0 \leq y \leq 2 - x$?
What are the absolute extrema of $f(x, y) = x^2y - y^3$ on the triangle bounded by $0 \leq x \leq 2$ and $0 \leq y \leq 2 - x$?
What is the equation of the surface modeled by $z = 25 - 2x^2 - 4y^2$?
What is the equation of the surface modeled by $z = 25 - 2x^2 - 4y^2$?
Which of the following equations represents the xy-trace of the surface modeled by the equation $y^2 + 4x^2 - 2z^2 = 1$?
Which of the following equations represents the xy-trace of the surface modeled by the equation $y^2 + 4x^2 - 2z^2 = 1$?
Which of the following equations represents the xz-trace of the surface modeled by the equation $y^2 + 4x^2 - 2z^2 = 1$?
Which of the following equations represents the xz-trace of the surface modeled by the equation $y^2 + 4x^2 - 2z^2 = 1$?
Which of the following equations represents the yz-trace of the surface modeled by the equation $y^2 + 4x^2 - 2z^2 = 1$?
Which of the following equations represents the yz-trace of the surface modeled by the equation $y^2 + 4x^2 - 2z^2 = 1$?
What is the value of $y^3 + x^3$ when evaluated along the path $y = mx$, where $m > 0$?
What is the value of $y^3 + x^3$ when evaluated along the path $y = mx$, where $m > 0$?
Which of the following equations represents the surface modeled by the equation $z = 25 - 2x^2 - 4y^2$?
Which of the following equations represents the surface modeled by the equation $z = 25 - 2x^2 - 4y^2$?
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Study Notes
Surface Equations
- The equation $y^2 + 4x^2 - 2z^2 = 1$ represents a surface.
- The equation $z = 25 - 2x^2 - 4y^2$ represents an elliptic paraboloid.
Traces of the Surface
- The xy-trace of the surface $y^2 + 4x^2 - 2z^2 = 1$ is represented by $y^2 + 4x^2 = 1$.
- The xz-trace of the surface $y^2 + 4x^2 - 2z^2 = 1$ is represented by $4x^2 - 2z^2 = 1$.
- The yz-trace of the surface $y^2 + 4x^2 - 2z^2 = 1$ is represented by $y^2 - 2z^2 = 1$.
Path Evaluations
- The value of $y^3 + x^3$ when evaluated along the path $y = mx$, where $m > 0$, is $m^3x^3 + x^3$.
Function Approximations
- The linear approximation of $f(0.8, 0.8)$ for the function $f(x, y) = 4\cos(2x - y)$ at $(\pi/4, \pi/4)$ can be found using the linear approximation formula.
Extrema of a Function
- The absolute extrema of $f(x, y) = x^2y - y^3$ on the triangle bounded by $0 \leq x \leq 2$ and $0 \leq y \leq 2 - x$ can be found using the method of Lagrange multipliers.
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