Geometry in 3D Space Quiz

Calculates the distance between two points in three-dimensional space.

Finds the coordinates of the midpoint of a line segment connecting two points in 3D.

Angles formed by a line and positive x, y, and z axes.

Cosines of the direction angles of a line.

Any multiple of direction cosines, used to describe the direction of a line.

The length of the line segment joining two points in three-dimensional space

Finds the coordinates of the middle point of a line segment given its endpoints coordinates

Direction angles are always in the range of 0 to 180 degrees (or 0 to π radians).

To find a point's location, draw perpendicular lines from the point to the x-axis and y-axis. The x-coordinate is the distance from the origin to the x-axis intersection; the y-coordinate is the distance from the origin to the y-axis intersection.

The horizontal position of a point in a plane, measured from the origin in relation to the x-axis.

The vertical position of a point in a plane, measured from the origin in relation to the y-axis.

The point (0,0) where the x and y axes intersect.

A space defined by three mutually perpendicular axes (x, y, and z).

The eight regions in 3D space divided by the x, y, and z axes.

The plane where all points have an x-coordinate of zero; the y-z plane.

Three intersecting lines, forming perpendicular directions necessary to determine any point's location in space.

The limit of a multivariable function f(x, y) as (x, y) approaches (a, b) exists if the function approaches the same value regardless of how (x, y) approaches (a, b).

A limit of a multivariable function does not exist if different paths to a point produce different function values.

The different directions (straight lines or curves) used to investigate what happens as (x, y) approaches (a, b) in a multivariable limit.

The process of calculating values of a multivariable function and observing its behaviors as the inputs approach a certain point.

The procedure of determining if a multivariable limit exists and finding its value, often involving substitution and/or trigonometric substitution.

Demonstrating that approaching a point along different paths produces different limit values.

The approach of using substitution to find the value of a multivariable limit.

Converting x and y coordinates into polar coordinates (r, θ): x = rcos(θ) , y = rsin(θ) , and then evaluating the limit as r approaches zero.

The rate of change of a multivariable function with respect to one variable, holding all other variables constant.

The notation ∂z/∂x represents the partial derivative of z with respect to x, holding y constant.

The partial derivative ∂z/∂x represents the slope of the tangent line to the curve of intersection of the surface z = f(x, y) and the plane y = constant. Similarly, ∂z/∂y represents the slope of the tangent line to the curve of intersection of the surface and the plane x = constant.

The partial derivatives of the first-order partial derivatives (e.g., ∂²z/∂x² or ∂²z/∂y∂x)

A partial derivative where you differentiate with respect to different variables in succession (e.g., ∂²z/∂y∂x).

When taking a partial derivative, all variables except the one you are differentiating with respect to are treated as constants.

Example: For z = x² + 2xy, ∂z/∂x = 2x + 2y (holding y constant), and ∂z/∂y = 2x (holding x constant).

Partial derivatives are used extensively in multivariable calculus, optimization, and engineering, for understanding how the output of a system changes when one input changes.

The rate of change of a multivariable function (z=f(x,y)) with respect to x, keeping y constant.

The rate of change of a multivariable function (z=f(x,y)) with respect to y, keeping x constant.

Treat y as a constant and differentiate z with respect to x using regular differentiation rules.

Treat x as a constant and differentiate z with respect to y using regular differentiation rules.

Represents the partial derivative of z with respect to x. Can also be written as fx or ∂f/∂x.

Represents the partial derivative of z with respect to y. Can also be written as fy or ∂f/∂y.

∂z/∂x = (2x)/(x^2 + y^2) - 1/(x+y)

∂z/∂x = -5x^4 y^4 sin(x^5 y^4)

∂z/∂x = x^4 y^3 cos(xy^3) + 4x^3 sin(xy^3)

∂w/∂x = 2x - yz

A partial derivative represents the rate of change of a multivariable function with respect to one variable, holding all other variables constant. Geometrically, it measures the slope of the tangent line to the function's graph along a specific direction.

Euler's theorem states that if a function f(x, y) and its partial derivatives (fx, fy, fxy, fyx) are continuous in an open region, then the mixed partial derivatives fxy and fyx are equal at any point within that region. In other words, the order of differentiation doesn't matter.

The chain rule extends to partial differentiation, allowing us to calculate the derivative of a composite function where the inner function depends on multiple variables. Basically, you multiply the derivative of the outer function with respect to the inner function by the derivative of the inner function with respect to the independent variable.

The order of differentiation in mixed partial derivatives refers to the sequence in which you differentiate a function with respect to different variables. For example, ∂²f/∂x∂y means differentiating with respect to y first, then x.

A function is continuous if its graph can be drawn without lifting the pen. In other words, there are no breaks or jumps in the graph. This is important for Euler's Theorem and mixed partial derivatives, as they require continuity for the order of differentiation to be interchangeable.

A partial derivative of order n is obtained by differentiating a function n times with respect to one or more independent variables. For example, ∂³f/∂x²∂y is a third-order partial derivative, meaning it involves three differentiations, two with respect to x and one with respect to y.