∫ calculus

1. **State the problem:** We want to design an open rectangular box with a square base of side length $b$ cm and height $h$ cm such that the volume is fixed at 256 cm³. We need to

1. **Problem 1: Find one-sided and two-sided limits from the graph of $y=f(x)$** Given the graph with vertical asymptote at $x=2$ and horizontal asymptote at $y=-2$, hollow circles

1. **State the problem:** Find the limit $$\lim_{x \to 2} \frac{x^2 - 4}{\sqrt[3]{x - 3} + 1}$$. 2. **Recall the formula and rules:** The limit of a function as $x$ approaches a va

1. The problem is to find the limit: $$\lim_{x \to 2} \frac{3}{x-3} + 1$$. 2. Recall that the limit of a sum is the sum of the limits, provided the limits exist.

1. Problem: Use the definition of the derivative to find $f'(x)$ for $f(x) = a^x$. 2. The definition of the derivative is:

1. Problem: Find the derivative $\frac{dy}{dx}$ of the function $y = x^x \cosh^2(x^3)$.\n\n2. Formula and rules: \n- Use the product rule: $\frac{d}{dx}[u v] = u' v + u v'$.\n- Use

1. The problem is to understand and apply the concept of linear approximation using derivatives. 2. Linear approximation uses the tangent line at a point to approximate the value o

1. **Problem a:** Find $\frac{dy}{dx}$ for $y = \sqrt{x^2 + 2}$. - Rewrite $y$ as $y = (x^2 + 2)^{1/2}$.

1. **Stating the problem:** Find the derivative of the function $y = 3x^2 + 5x - 7$. 2. **Formula used:** The derivative of a function $f(x)$ with respect to $x$ is given by $f'(x)

1. השאלה היא למצוא את נקודות הקיצון של פונקציה נתונה. 2. נקודות קיצון הן נקודות שבהן הנגזרת הראשונה של הפונקציה שווה לאפס, כלומר $$f'(x)=0$$, והנגזרת השנייה בודקת אם זו נקודת מינימ

1. נתחיל מהגדרת הפונקציה: $$f(x) = x^2 e^x - 1$$. 2. נמצא את הנגזרת של הפונקציה כדי להבין את השיפועים והנקודות הקריטיות. נשתמש בכלל המכפלה: $\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(

1. **Stating the problem:** We want to understand the meaning of the second derivative $y''$ in calculus. 2. **Formula and explanation:** The first derivative $y' = \frac{dy}{dx}$

1. **State the problem:** We want to find the limit as $x \to 0$ of the expression $$\frac{1 + \sin x - \cos x}{1 + \sin(px) - \cos(px)}.$$\n\n2. **Recall the relevant formulas and

1. The problem is to determine when to use the first derivative or the second derivative in calculus. 2. The first derivative of a function $f(x)$, denoted as $f'(x)$, represents t

1. **Problem:** Find the derivative $\frac{dy}{dx}$ of the function $$y = e^{\cos x} \times \sin^3(x)$$ 2. **Formula and rules:**

1. Let's start by stating the problem: Find the integral of trigonometric ratios such as $\sin x$, $\cos x$, $\tan x$, $\cot x$, $\sec x$, and $\csc x$. 2. The general formula for

1. The problem asks about the indeterminate form in a given case. 2. Indeterminate forms occur in limits when substitution leads to expressions like $\frac{0}{0}$, $\frac{\infty}{\

1. **State the problem:** Find the limit $$\lim_{x \to 7} \frac{x^2 - 6x - 7}{2x^2 - 5x - 63}$$. 2. **Recall the limit rule:** If direct substitution leads to an indeterminate form

1. **State the problem:** Find the derivative of the function $$f(x) = 1 - \frac{\ln((x-1)^2)}{x-1}$$. 2. **Recall the formulas:**

1. The problem is to find the integral of a function, but since the function is not specified, let's consider a general example: find $\int x^2 \, dx$. 2. The formula for integrati

1. The problem is to investigate the sign of the derivative of a function, which helps us understand where the function is increasing or decreasing. 2. The derivative of a function