The derivative of a polynomial is obtained by differentiating each term of the polynomial with respect to the variable. For example, let’s consider the polynomial: f(x) = 5x^3 + 2x^2 – 7x + 4 To find its derivative, we differentiate each term with respect to x: f'(x) = (d/dx)(5x^3) + (d/dx)(2x^2) – (d/dx)(7x) + (d/dx)(4)…
The intermediate value property is a property of continuous functions that states that if a continuous function f(x) takes on two values, say a and b, at two different points a and b in its domain, then it must take on every value between a and b at some point c in its domain. Formally,…
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…
Continuity is a fundamental concept in calculus that describes how a function behaves at every point in its domain. A function is said to be continuous if it has no abrupt jumps, breaks, or holes in its graph. More formally, a function f(x) is continuous at a point x=a if three conditions are met: If…
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.…
The angle between a line and a plane is the angle formed by the intersection of the line and the plane. This angle is measured as the acute angle between the line and the plane, i.e., the smallest angle between them. To calculate the angle between a line and a plane, you can use the…
Skew lines are two non-intersecting lines that are not parallel to each other. In other words, skew lines are two lines in three-dimensional space that do not lie in the same plane and do not intersect each other. Skew lines are important in geometry and can be used to solve various problems, such as finding…
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…
The standard equation of a parabola is: y = a x^2 + b x + c where: The vertex form of a parabola is: y = a(x – h)^2 + k where: The focus and directrix of a parabola can also be expressed in terms of a, as follows: Note that the vertex form of…
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…
