September 4, 2024
To find the integral of x 2, written as \(\displaystyle \int x^2 \, dx\), we use the Power Rule for integration. The Power Rule for differentiation states: \[ \dfrac{d}{dx}(x^n) = n \cdot x^{n-1} \] To integrate, we reverse this process with what’s called the Reverse Power Rule: \[ \displaystyle \int x^n \, dx = \frac{x^{n+1}}{n+1} […]
September 4, 2024
Have you ever wondered how to find the integral of \(\sqrt{x}\)? It’s actually quite simple once you know the power rule! Let’s start by rewriting \(\sqrt{x}\) in a way that’s easier to work with. We know that: \[ \sqrt{x} = x^{\frac{1}{2}} \] Now, we can use the power rule for integration. The power rule says […]
September 4, 2024
Introduction In calculus, we often need to find the limit of a function as it approaches a certain value. A common scenario involves the limit of a ratio of two functions, which we call rational functions. The general form looks like this: $$\lim_{x \to a} \frac{f(x)}{g(x)}$$ This expression asks: “What value does the fraction approach […]
September 2, 2024
Series are one of the most frustrating topics in calculus. They can be very confusing, and it can be very hard to determine if a series converges or not. If that weren’t enough, there are two types of convergence – absolute and conditional. This article will take you through what conditional convergence is and how […]
September 2, 2024
This article will teach you how to take the derivative of $4e^x$, as well as the proof for why the derivative is correct. Taking the Derivative Let us take the derivative of $\ln e^x$. Of course, $\ln e^x = x$, so the derivative should equal $1$. Using the chain rule, we get that $\frac{d}{dx} \ln […]
September 2, 2024
To find the derivative of 1, we will start by graphing the equation $y = 1$, pictured below: The derivative of a function at a point is the rate of change of a function at said specific point. In other words, it is the slope of the tangent line to the curve at a […]
August 19, 2024
The Ratio Test is a very nice tool to use for determining whether or not a series converges, but it can get a bit complicated. Here it is: If we have a series \(\sum a_n\), let there be a number \(L = \lim_{n\to\infty} \left|\dfrac{a_{n+1}}{a_n}\right|\). – If \(L < 1\), the series converges absolutely. – If […]
August 19, 2024
Because an integral is, in essence, the inverse function to a derivative, the integral of \(\cos x\) is going to be the function whose derivative is equal to \(\cos x\). To do this, let’s look at the derivatives for the basic trigonometric functions: \[ \frac{d}{dx}(\sin x) = \cos x \] \[ \frac{d}{dx}(\cos x) = -\sin […]
August 19, 2024
Have you ever wondered how to take the derivative of an exponential function like \(2^x\) or \(4^{3x^2}\)? In this article, we will derive the formula for taking the derivative of any exponential function whatsoever. Derivation of the Formula In order to derive the formula for the derivative of an exponential function, we first need to […]
