The value of \(\cos 180^\circ\) is: $\cos 180^\circ = \boxed{-1}$ Explanation To understand why, let’s look at the unit circle. The cosine of an angle corresponds to the \(x\)-coordinate of the point on the unit circle at that angle. We measure angles by moving counterclockwise from the positive \(x\)-axis. Thus at \(180^\circ\), we’re at the…
Introduction The limit of an absolute value function often involves determining how the function behaves as the input approaches a particular point, especially around points where the expression inside the absolute value changes sign. Key Concept Often limits involving absolute value do not exist. For example: $$\lim_{x\to0}\dfrac{x}{|x|}.$$ Because of the discontinuity on the graph of…
1 Introduction Oftentimes in calculus, we must compute limits that involve trig functions. For example, $$\lim_{x\to{a}}\sin{x}$$ or the limit of any other expression that contains a trig function. This specific limit asks “what value does $\sin{x}$ approach as $x$ approaches $a$?” Keep in mind that some trig limits do not exist. Because of the oscillation…
Introduction One of the most common related rates problems is the ladder problem, which looks at how a ladder slides down a wall, assuming that the ladder always makes a right triangle with the wall. Let’s see how to solve these sorts of problems by working through a simple example. Example Let’s consider a ladder…
Polar equations can seem tricky, especially when it comes to taking derivatives. However, the derivative of a polar equation can be taken directly in polar coordinates without needing to convert to rectangular form. This article will guide you through the process with helpful examples. Which Derivative Are You Computing? In polar coordinates, you may want…
Basic Derivative Formulas Constant Rule: $\dfrac{d}{dx}C=0$ Constant Multiple Rule: $\dfrac{d}{dx}\Big(Cf(x)\Big)=C\dfrac{d}{dx}f(x)$ Sum and Difference Rule: $\dfrac{d}{dx}\Big(f(x)\pm g(x)\Big)=\dfrac{d}{dx}f(x)\pm\dfrac{d}{dx}g(x)$ Power Rule: $\dfrac{d}{dx}x^n=nx^{n-1}$ Product Rule: $$\dfrac{d}{dx}\Big(f(x)g(x)\Big)=f(x)g'(x)+g(x)f'(x)$$ Quotient Rule: $$\dfrac{d}{dx}\left(\dfrac{f(x)}{g(x)}\right)=\dfrac{g(x)f'(x)-f(x)g'(x)}{g(x)^2}$$ Chain Rule: $$\dfrac{d}{dx}\Big(f(g(x))\Big)=f'(g(x))g'(x)$$ Where $f(x)$ is the outside function and $g(x)$ is the inside function. Exponential and Logarithmic Derivatives \[ \begin{array}{lll} \dfrac{d}{dx}e^x=e^x && \dfrac{d}{dx}a^x=a^x\ln{a} \\[10pt] \dfrac{d}{dx}\ln{x}=\dfrac{1}{x} && \dfrac{d}{dx}\log_a x=\dfrac{1}{x\ln{a}} \end{array}…
This article will walk you through how to prove the Power Rule and take the derivative of any polynomial. We are going to prove every step of the way. If you don’t want to see the proof and just want a formula, then just know that for any function $f(x) = x^n$, $f'(x) = nx^{n-1}$;…
Fundamental Theorem of Calculus: Part one: $$\int_a^bf(x)\:dx=F(b)-F(a)$$ Where $F(x)$ is an antiderivative of $f(x)$ Part two: $$\dfrac{d}{dx}\int^x_af(t)\:dt=f(x)$$ Basic Integration Rules: Constant Rule: $\displaystyle{\int} 0 \:dx =C$ Constant Multiple Rule: $\displaystyle{\int} Cf(x) \:dx = C\displaystyle{\int}f(x) \:dx$ Sum and Difference Rule: $\displaystyle{\int}\left[ f(x)\pm g(x) \right]\:dx = \displaystyle{\int}f(x) \:dx\pm \displaystyle{\int}g(x)\:dx$ Power Rule: $\displaystyle{\int}x^n \:dx= \dfrac{x^{n+1}}{n+1}+C$ Inverse Bounds Rule:…
Hello everyone! In this blog, we will be reviewing everything about derivatives that you need to know for the AP Calculus test, from the definition of a derivative to more advanced topics like implicit differentiation. This blog is for the curious who want to see the proofs of the differentiation rules we all use. Without…
Hello everyone! In this article, we will go over how to take the integral of $\dfrac{x-1}{x+1}$. The technique shown in this article can also be used for integrating many fractions of rational equations, so it’s definitely a handy tool to know. Taking the Integral of x-1 over x+1 We will start expressing $\dfrac{x-1}{x+1}$ in a…
