November 8, 2024
Introduction Integration is one of the two main operations in calculus, the other being differentiation. Integrals allow us to calculate areas, volumes, accumulated quantities, and more. To simplify integration, mathematicians have developed a set of integration rules. These rules make it easier to work with different types of functions and solve integrals efficiently. In this […]
November 8, 2024
Introduction The concept of an integral is a fundamental part of calculus and mathematical analysis, allowing us to measure quantities that accumulate over time or space. Integrals are used extensively in physics, engineering, economics, and many other fields to calculate areas, volumes, and total values of quantities. In this article, we’ll cover the basic definition […]
November 8, 2024
\[\begin{align*} \boxed{2^3 = 2 \times 2 \times 2 = 8} \end{align*}\] Introduction Exponents are a convenient way to represent repeated multiplication, and understanding how to evaluate them is essential in math. The expression \( 2^3 \) is an example of exponential notation, where \( 2 \) is the base, which is the number being multiplied, […]
November 8, 2024
Introduction Logarithms are the inverse operations of exponents, allowing us to “undo” exponentiation and solve equations involving exponential terms. We will cover the main rules of logarithms, which make it easier to manipulate logarithmic expressions. Definition of Logarithms A logarithm answers the question: “To what exponent must a base be raised to get a certain […]
November 8, 2024
Introduction Exponentiation is a notation in mathematics representing repeated multiplication. They provide a way to express large numbers more concisely and simplify calculations in algebra, calculus, and many applied fields. Definition of Exponents An exponent is a small number placed above and to the right of a base number, indicating how many times to multiply […]
November 8, 2024
Exponent laws/rules are useful tools when evaluating expressions. They include the product rule, quotient rule, and power rules. Given integers $m$ and $n$, we have the following: Product Rule \[\begin{align*} x^m \cdot x^n = x^{m + n} \end{align*}\] This works as if we expand the exponent out as multiplications of $x$, then $x^m = (x […]
November 7, 2024
\[\begin{align*} \boxed{ e^{\ln x} = x } \end{align*}\] What is $e$? \( e \) is a special number in math, approximately equal to 2.718. It is also called Euler’s constant and is the base of a logarithm called the natural logarithm. You can think of it as the number that makes many things in math […]
November 1, 2024
\[\begin{align*} \boxed{\int 2^x \, dx = \frac{2^x}{\ln(2)} + C} \end{align*}\] where \( C \) is the constant of integration. Step-by-Step Solution: \[\begin{align*} \int 2^x \, dx \end{align*}\] Since \( 2^x \) is an exponential function with a base other than \( e \), to simplify the integration process, we can rewrite \( 2^x \) in […]
November 1, 2024
\[\begin{align*} \boxed{\frac{d}{dx} \left( 2^x \right) = 2^x \ln(2)} \end{align*}\] Introduction: Finding the derivative of \( 2^x \) helps us understand how the function changes as \( x \) changes. Unlike derivatives of polynomials or basic trigonometric functions, differentiating an exponential function involves the natural logarithm. Let’s walk through the steps. Step-by-Step Solution: To differentiate \( […]
