Integration by substitution is a technique used to simplify an integral by replacing the variable of integration with a new variable. This new variable is chosen so that the resulting integral becomes easier to evaluate. The basic steps for integration by substitution are as follows: Partial fraction decomposition is a technique used to simplify a…
The fundamental theorem of calculus is a theorem that links the concept of differentiation with that of integration, and it has two parts. Part 1: Let f be a continuous function on the interval [a,b], and let F be a function defined by F(x) = ∫a^x f(t) dt where a ≤ x ≤ b. Then…
A definite integral is a mathematical concept that represents the area under a curve between two specified points on the x-axis. It is denoted by the symbol ∫, and is defined as: ∫[a,b] f(x) dx = F(b) – F(a) where f(x) is a function, F(x) is its antiderivative or indefinite integral, and a and b…
When dealing with implicit functions, we often have equations of the form: F(x, y) = 0 where y is an implicit function of x. That is, we can’t solve for y explicitly in terms of x. However, we can still differentiate the equation with respect to x to find the derivative of y with respect…
Rolle’s Theorem and Lagrange’s Mean Value Theorem are two important results in calculus that relate to the behavior of functions on a given interval. Rolle’s Theorem states that if a function f(x) is continuous on the closed interval [a, b], differentiable on the open interval (a, b), and f(a) = f(b), then there exists at…
The derivative of a function f(x) gives the rate of change of f(x) with respect to x. The derivative of order two, or the second derivative of f(x), represents the rate of change of the first derivative of f(x) with respect to x. Mathematically, the second derivative of f(x) is denoted as f”(x) or d^2/dx^2…
In differential calculus, the concept of tangents and normals is closely related to the idea of derivatives. The derivative of a function at a point gives the slope of the tangent line to the curve at that point. The derivative is also used to find the equation of the tangent line to the curve at…
Exponential and logarithmic functions are two important types of mathematical functions commonly used in many fields, including mathematics, physics, economics, and engineering. Exponential functions have the form f(x) = a^x, where a is a positive constant called the base of the function. These functions have the property that the value of the function increases or…
Differential calculus is a branch of calculus that deals with the study of rates of change and slopes of curves. Trigonometry is a branch of mathematics that deals with the study of triangles and the relationships between their sides and angles. The two subjects are related in that trigonometric functions, such as sine, cosine, and…
Rationality refers to the ability to think logically, make sound judgments, and make decisions based on reason rather than emotions or impulses. It involves using critical thinking skills to analyze information, evaluate evidence, and draw conclusions based on facts and evidence. In general, a rational person is someone who can separate their emotions from their…
