To solve the problem, you need to start by designing an interface for functions. Although using named local classes such as “Sinus” and “Square” would be the conventional approach, you can also opt for anonymous local classes if you prefer. Question: Greetings everyone, I would be grateful if someone could provide me with solutions along […]

# Category: Calculus

## Evaluating the limit of (x, y) approaching (0, 0) for the expression xy divided by the square root of x squared plus y squared

Question: I’m attempting to evaluate the limit $$ lim_{(x,y)to (0,0)} frac{x-sqrt{xy}}{x^2-y^2} $$ over the domain $x>0$ , $y>0$ . =========== My attempt: $f(x,x^2)to +infty$ ; therefore, if the limit exists, it must be $+infty$ . Hint: Take note that, $$|x|=sqrt{x^2}leq sqrt{x^2+y^2}, $$ and the same applies to $|y|$. Solution 2: By transforming to polar coordinates, […]

## Discovering Graph-Based Methods for Determining Velocity

Displacement is the overall distance traveled in a specific direction, while distance refers to the overall distance without considering direction. For instance, if someone walks 3 feet north, 3 feet east, 3 feet south, and 3 feet west, their net distance from the starting point is 0. In this case, the direction could be considered […]

## Utilize l’Hospital’s rule to determine the limit

Question: Here is the limit I am attempting to evaluate: $$ limlimits_{x to infty} frac{x^2 + mathrm{e}^{4x}}{2x- mathrm{e}^x} $$ Now, initially, I am trying to determine the indeterminate form in order to apply L’Hospital’s rule. However, when I differentiate the numerator and denominator using L’Hospital’s rule and simplify, I end up with the same form […]

## Understanding the derivative in polar coordinates

The following equation is speculative, at least: $$begin{bmatrix}dfrac{partial z}{partial r} & dfrac{partial z}{partialtheta}\dfrac{partialoverline{z}}{partial r} & dfrac{partialoverline{z}}{partialtheta}end{bmatrix}begin{bmatrix}dfrac{partial r}{partial z} & dfrac{partial r}{partialoverline{z}}\dfrac{partialtheta}{partial z} & dfrac{partialtheta}{partialoverline{z}}end{bmatrix}=begin{bmatrix}1 & 0\0 & 1end{bmatrix}$$ First, we calculate $dfrac{partialoverline{z}}{partial r}$ and $dfrac{partialoverline{z}}{partialtheta}$ using the equations $dfrac{partialoverline{z}}{partial r}=costheta-isintheta=e^{-itheta}$ and $dfrac{partialoverline{z}}{partialtheta}=-rsintheta-ircostheta=rsinleft(-thetaright)-ircosleft(-thetaright)=-left(-rsinleft(-thetaright)+ircosleft(-thetaright)right)=-ire^{-itheta}$. We can then plug these values into the equation to get: $$begin{bmatrix}e^{itheta} & […]

## Curvature-induced surface area through revolution

Hence, we can conclude from this step that the equation for ds is given by $$ds = sqrt{(dfrac{dr}{dtheta})^2+r^2} dtheta$$. When deriving the surface of revolution (using frustrums, not shown here), we should note that just as multiplying the arc length of a curve by $2pi * left(x(t) ,mathrm{or},y(t)right)$ is required to obtain its surface area […]

## Algebraic Method for Determining One-Sided Limits without Relying on Graphs

The function f(x), which represents the amount of drug in the bloodstream after t hours, is shown in the graph. The equation for f(x) is given by f(x) = x/(x^2-1), which can be further simplified as x/((x-1)(x+1)). It is important to note that there are vertical asymptotes located at x=1 and x=-1. Question: Is there […]

## Clarifying the Difference Between Gradient and Directional Derivative

Calculate the gradient of $f$ at the point (0,2,4). Here are the steps I’ve taken so far: First, I normalized the vector (-2,1,2) to obtain (-2/3, 1/3, 2/3). We can express $f$ as $f(x,y,x^2+y^2)$ since x=0, y=2, and $x^2+y^2 = 4$. Therefore, $f(0,2,4) = 2 cdot 0$. Regarding Clarifying Question 2, the magnitude of vector […]

## Duplicate: Determining the Limit of $limlimits_{x→∞}frac{sin x}{x}$

For any $|x|lt 1$, the limit of $x^n$ as $n$ approaches infinity is equal to zero. Solution 2 involves the limit where $h$ approaches zero, resulting in an indeterminate form of zero over zero. However, I require a comprehensive formal proof that utilizes the definition of limit ($ε, δ$) and does not rely on techniques […]

## Comparison of Newton and Leibniz Notation

Leibniz eventually realized that the current notation is more accurate and provides better intuition for the chain rule, among other things. Newton’s ideas were valuable in the fields of physics and mechanics, which is fitting given his background as a physicist. Prof. Arnold distinguished between Newton and Leibniz’s approaches to mathematical analysis, with the latter […]