The continuity of composite functions is governed by the following theorem: Let f be a function defined on an interval I containing a point a, and let g be a function defined on an interval J containing f(a). If f is continuous at a and g is continuous at f(a), then the composite function g…
Let $f(x)$ and $g(x)$ be two functions, and let $c$ be a real number. The sum of the two functions $f(x)$ and $g(x)$ is defined by $(f+g)(x) = f(x) + g(x)$ for all $x$ in the domain of both functions. To determine if the sum of $f(x)$ and $g(x)$ is continuous at $c$, we need…
The limit of a function at a real number is a fundamental concept in calculus that describes the behavior of a function as the input values approach a specific real number. Formally, we say that the limit of a function f(x) as x approaches a real number c is L if for every ε >…
Coplanar lines are lines that lie on the same plane. In other words, any two points on each of the lines can be connected with a straight line that lies entirely in the same plane. For example, if two lines intersect, they are coplanar because the plane containing one line also contains the other line.…
Direction cosines and direction ratios are used to describe the orientation of a line or a vector in three-dimensional space. Direction cosines are the cosines of the angles that a given line or vector makes with the positive x, y, and z axes of a Cartesian coordinate system. For example, if the angles that a…
The distance between two points in three-dimensional space can be found using the following formula: d = √((x2 – x1)^2 + (y2 – y1)^2 + (z2 – z1)^2) where (x1, y1, z1) and (x2, y2, z2) are the coordinates of the two points, and d is the distance between them. To use the formula, you…
To find the equation of a circle passing through the points of intersection of two circles, we can follow these steps: Let’s say the two circles have equations: (x – a)^2 + (y – b)^2 = r^2 (x – c)^2 + (y – d)^2 = s^2 where (a,b) and (c,d) are the centers of the…
Circumscribed circle In calculation, the encompassed circle or circumcircle of a polygon is a circle that goes through all the vertices of the polygon. The focal point of this circle is known as the circumcenter and its range is known as the circumradius. Few out of every odd polygon has an encompassed circle. A polygon…
In calculation, an orthocentric framework is a bunch of four focuses on a plane, one of which is the orthocenter of the triangle shaped by the other three. Proportionately, the lines going through disjoint matches among the focuses are opposite, and the four circles going through any three of the four focuses have a similar…
In analytical geometry, the centroid of a plane figure is the point where its medians intersect. A median is a line segment connecting a vertex of the figure to the midpoint of the opposite side. The centroid is often referred to as the “center of mass” or “center of gravity” of the figure, as it…
