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Questions and Answers
What is the result of the expression $2x - x^2$?
What is the result of the expression $2x - x^2$?
- $x^2 - 2x$
- $x(x - 2)$ (correct)
- Cannot be determined
- $x(x + 2)$
What are the solutions to the equation $y^2 = 2x + 6$ and $y = x - 1$?
What are the solutions to the equation $y^2 = 2x + 6$ and $y = x - 1$?
- $(5, -3)$ and $(4, -2)$
- $(1, 1)$ and $(2, 2)$
- $(0, 0)$ and $(2, 4)$
- $(5, 4)$ and $(-1, -2)$ (correct)
What is the value of $x$ when $y = 2x^2$ and $y = x^2 + 1$?
What is the value of $x$ when $y = 2x^2$ and $y = x^2 + 1$?
- $0$ and $1$
- $-1$ and $1$ (correct)
- $-1$ and $0$
- $1$ and $2$
Which statement represents the area of the ellipse using double integrals?
Which statement represents the area of the ellipse using double integrals?
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What is the result of the expression $x^2 - y^2$ when using polar coordinates?
What is the result of the expression $x^2 - y^2$ when using polar coordinates?
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Study Notes
Algebraic Expressions
- The expression $2x - x^2$ is a quadratic polynomial.
Systems of Equations
- The equation $y^2 = 2x + 6$ is a quadratic equation in $y$.
- The equation $y = x - 1$ is a linear equation in $y$.
- Solving these equations simultaneously involves finding the intersection points.
Quadratic Equations
- The equation $y = 2x^2$ is a quadratic equation in $x$.
- The equation $y = x^2 + 1$ is a quadratic equation in $x$.
- Solving these equations simultaneously involves finding the values of $x$ that satisfy both equations.
Integration and Coordinate Systems
- The area of an ellipse can be represented using double integrals.
- The expression $x^2 - y^2$ can be converted to polar coordinates using the substitution $x = r\cos(\theta)$ and $y = r\sin(\theta)$.
- This substitution allows for the evaluation of the integral in polar coordinates.
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