Vector Valued Functions

Front

Define a vector-valued function.

Back

A vector-valued function is a function that takes one or more variables and returns a vector. It is often expressed in terms of its components, such as \( \mathbf{r}(t) = \langle x(t), y(t), z(t) \rangle \).

Front

What is the geometric interpretation of a vector-valued function in 3D space?

Back

The geometric interpretation of a vector-valued function in 3D space is that it describes a curve or path traced out by the function as the parameter varies.

Front

What is the parametric representation of a curve?

Back

A parametric representation of a curve expresses the coordinates of the points on the curve as functions of a parameter, typically denoted as \( t \). For example, \( x(t) \) and \( y(t) \) define the curve in the xy-plane.

Front

What is the difference between a scalar function and a vector-valued function?

Back

A scalar function maps inputs to a single real number, while a vector-valued function maps inputs to a vector in a multi-dimensional space.

Front

How do you find the derivative of a vector-valued function?

Back

To find the derivative of a vector-valued function \( \mathbf{r}(t) = \langle x(t), y(t), z(t) \rangle \), differentiate each component with respect to \( t \): \( \mathbf{r}'(t) = \langle x'(t), y'(t), z'(t) \rangle \).

Front

What is the arc length of a curve defined by a vector-valued function?

Back

The arc length \( L \) of a curve defined by a vector-valued function \( \mathbf{r}(t) \) from \( t=a \) to \( t=b \) is given by the integral: \( L = \int_a^b ||\mathbf{r}'(t)|| dt \).

Front

What is a quadric surface?

Back

A quadric surface is a second-degree algebraic surface in three-dimensional space, defined by a quadratic equation in three variables, such as \( Ax^2 + By^2 + Cz^2 + Dxy + Exz + Fyz + Gx + Hy + Iz + J = 0 \).

Front

What type of quadric surface is represented by the equation \( z = x^2 + y^2 \)?

Back

The equation \( z = x^2 + y^2 \) represents a paraboloid, specifically an elliptic paraboloid.

Front

What is the equation of a cylinder in vector form?

Back

The equation of a cylinder in vector form can be expressed as \( \mathbf{r}(t) = \langle r \cos(t), r \sin(t), z \rangle \), where \( r \) is the radius and \( z \) is the height.

Front

How do you determine the type of quadric surface from a vector-valued function?

Back

To determine the type of quadric surface from a vector-valued function, analyze the coefficients of the quadratic terms in the equation derived from the function and identify the surface type based on standard forms.