10 Questions
Which equation represents the surface modeled by the equation $y^2 + 4x^2 - 2z^2 = 1$?
$z = 1 - 2x^2 - y^2$
What is the value of $y^3 + x^3$ when evaluated along the path $y = mx$, where $m > 0$?
$m^3$
What is the linear approximation of $f(0.8, 0.8)$ for the function $f(x, y) = 4\cos(2x - y)$ at $(\pi/4, \pi/4)$?
$2\sqrt{2}$
What are the absolute extrema of $f(x, y) = x^2y - y^3$ on the triangle bounded by $0 \leq x \leq 2$ and $0 \leq y \leq 2 - x$?
Maximum at $(2, 0)$ and minimum at $(0, 2)$
What is the equation of the surface modeled by $z = 25 - 2x^2 - 4y^2$?
A paraboloid
Which of the following equations represents the xy-trace of the surface modeled by the equation $y^2 + 4x^2 - 2z^2 = 1$?
$y = \sqrt{1 - 4x^2 + 2z^2}$
Which of the following equations represents the xz-trace of the surface modeled by the equation $y^2 + 4x^2 - 2z^2 = 1$?
$x = \sqrt{1 - 4y^2 + 2z^2}$
Which of the following equations represents the yz-trace of the surface modeled by the equation $y^2 + 4x^2 - 2z^2 = 1$?
$z = \sqrt{1 - 4x^2 + 2y^2}$
What is the value of $y^3 + x^3$ when evaluated along the path $y = mx$, where $m > 0$?
$m^3x^3$
Which of the following equations represents the surface modeled by the equation $z = 25 - 2x^2 - 4y^2$?
$x^2 + 4y^2 - 2z^2 = 25$
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.
This quiz focuses on topics related to Calculus III, specifically Chapter 14. It includes examples and practice problems related to writing equations, sketching traces, evaluating expressions, and estimating values using linear approximation. Perfect for students studying Calculus at Loyola Marymount University in Fall 2023.
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