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Parametric equation of a circle, center (0,0), radius r?
$x = r \cos(t)$, $y = r \sin(t)$
Magnitude of vector v = <a, b>?
$||v|| = \sqrt{a^2 + b^2}$
Position vector r(t), how to find velocity vector v(t)?
$v(t) = r'(t) = <\frac{dx}{dt}, \frac{dy}{dt}>$
Velocity vector v(t), how to find speed?
$Speed = ||v(t)|| = \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2}$
Position vector r(t), how to find acceleration vector a(t)?
$a(t) = v'(t) = r''(t) = <\frac{d^2x}{dt^2}, \frac{d^2y}{dt^2}>$
Determinant of a 2x2 matrix [[a, b], [c, d]]?
$det([[a, b], [c, d]]) = ad - bc$
Equation of a circle in standard form?
$(x-h)^2 + (y-k)^2 = r^2$
Standard form equation of an ellipse?
$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$
Standard form equation of a hyperbola?
$\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$ or $\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1$
Parametric form for x and y?
$x = f(t), y = g(t)$
Define Parametric Functions.
Functions where x and y coordinates are defined in terms of a parameter, often 't'.
Define Implicitly Defined Functions.
Functions where the relationship between x and y is defined by an equation, not explicitly solved for y.
Define Vectors.
Mathematical objects with both magnitude and direction, often represented as arrows.
Define Matrices.
Rectangular arrays of numbers used to represent linear transformations and solve systems of equations.
Define Conic Sections.
Shapes formed by slicing a cone: circles, ellipses, parabolas, and hyperbolas.
Define Vector-Valued Functions.
Functions that output vectors instead of scalar values, often used to describe the position of a moving object over time.
What is the magnitude of a vector?
The length of the vector.
What is the determinant of a matrix?
A scalar value that indicates if a matrix is invertible.
What is the dot product of two vectors?
A scalar value obtained by multiplying corresponding components of two vectors and summing the results.
What is the cross product of two vectors?
A vector that is perpendicular to both input vectors (defined in 3D space).
Explain Parametric Functions.
Functions that define x and y coordinates using a parameter, often 't', allowing representation of motion along a path.
Explain Implicitly Defined Functions.
Functions defined by a relationship between x and y, not explicitly solved for y, such as conic sections.
Explain Vectors.
Mathematical objects with magnitude and direction, used to represent physical quantities like force, velocity, and displacement.
Explain Matrices.
Rectangular arrays of numbers used to represent linear transformations, solve systems of equations, and perform complex operations.
Explain Conic Sections.
Shapes formed by slicing a cone, including circles, ellipses, parabolas, and hyperbolas, all examples of implicitly defined functions.
Explain Vector-Valued Functions.
Functions that output vectors, used to describe the position of a moving object over time, essential for understanding motion in physics and engineering.
Explain the significance of the derivative of a parametric function.
The derivatives dx/dt and dy/dt represent the rates of change of x and y with respect to the parameter t, indicating speed and direction.
Explain how matrices can act as functions.
Matrices can transform vectors, which is crucial in linear algebra and computer graphics for operations like scaling, rotation, and translation.
Explain the concept of parametrization.
Expressing implicitly defined functions using parameters (like 't') to make graphing and analysis easier, especially for complex shapes.
What does the determinant of a matrix tell you?
The determinant indicates whether the matrix is invertible; a non-zero determinant means the matrix has an inverse.