∫ calculus

1. **Stating the problem:** We are given the cost function $$C(x,y) = e^y (32x - 16x^2 - 7) + 2y e^{-x^2 - y^2}$$ where $x$ and $y$ represent production quantities of components.

1. Problem: Find the derivative $\frac{dy}{dx}$ of $y = 2x^5 - \cos x$. 2. Formula: Use the power rule $\frac{d}{dx} x^n = nx^{n-1}$ and the derivative of cosine $\frac{d}{dx} \cos

1. مسئله را بیان می‌کنیم: باید حد $$\lim_{x \to 2} \frac{x - 2\sqrt{x}}{x^2 - 7x + 12}$$ را محاسبه کنیم. 2. ابتدا صورت و مخرج را بررسی می‌کنیم. اگر مستقیم جایگذاری کنیم:

1. مسئله را بیان می‌کنیم: حد $$\lim_{x \to -\infty} \frac{2x^5 + 4x^3 - 1}{4x^3 - 3x^5 + 5}$$ را می‌خواهیم پیدا کنیم. 2. برای حد‌های بی‌نهایت در کسرهای چندجمله‌ای، درجه‌های بالاتر

1. Soal: Tentukan turunan pertama dan nilai turunan pada titik tertentu. 2. Rumus turunan dasar yang digunakan:

1. **State the problem:** Find the limit $$\lim_{x \to 3} \frac{\sqrt{x^3 - x - 1} - \sqrt{x^3 - 7x + 7}}{\sqrt{x^3 - x - 1} - \sqrt{x^3 - 3x + 3}}.$$\n\n2. **Identify the form:**

1. **Problem:** Find the centroid of the area bounded by the curves $y = 2x + 1$, $xy = 7$, and the vertical line $x = 8$. 2. **Step 1: Understand the boundaries**

1. **Problem:** Evaluate the integral $$\iint_D \frac{xy}{\sqrt{x^2 + y^2}}\,dxdy$$ over the region $$D = \{(x,y) : x^2 + y^2 \leq 1, xy \leq 0\}$$. 2. **Step 1: Understand the reg

1. **Problem:** Evaluate the integral $$\iint_D \frac{xy}{\sqrt{x^2 + y^2}}\,dxdy$$ over the region $$D = \{(x,y) : x^2 + y^2 \leq 1, xy \leq 0\}$$. 2. **Step 1: Understand the reg

1. **Problem:** Calculate the integral $$\int \frac{x^4 - 3}{x^2} \, dx$$. 2. **Formula and rules:** Recall that dividing powers of $x$ means subtracting exponents: $$\frac{x^a}{x^

1. **Problem statement:** Find the derivative $f'(x)$ of the function $f(x) = x^3 + 2x^2 + 4x + 5$. 2. **Formula used:** The derivative of a function $f(x)$ is found by applying th

1. نبدأ بكتابة الدالة المعطاة: $$f(x) = \frac{x - 7}{x - 3}$$ 2. المطلوب هو إيجاد نهاية الدالة عندما يقترب $x$ من 3، أي $$\lim_{x \to 3} \frac{x - 7}{x - 3}$$

1. **State the problem:** We want to find the dimensions of an open-top rectangular box with volume 32 ft³ that uses the least amount of material (surface area). 2. **Define variab

1. **State the problem:** We have a function defined as $f(x) = x^3 - x^2 - 8x + 5$ for $x < a$, and we know $f$ is increasing on this domain. We need to find the largest possible

1. **Problem Statement:** Find the maxima, minima, and inflection points of the function $$f(x) = (x-12)^3$$. 2. **Formula and Rules:**

1. **State the problem:** Show that $$x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = 2u \log u$$ where $$\log u = \frac{x^3 + y^3}{3x + 4y}$$. 2. **Identify th

1. Problem: Find the derivative $\frac{dy}{dx}$ of $y = 5x^4$. 2. Formula: Use the power rule for derivatives, which states:

1. **State the problem:** Find the limit $$\lim_{x \to -3} \frac{4x^2 + 12x}{x^2 + 4x + 3}$$. 2. **Check direct substitution:** Substitute $x = -3$ into numerator and denominator.

1. We are asked to find the limit: $$\lim_{x \to 2} \frac{x^4 + 3x^3 - 10x^2}{x^2 - 2x}$$. 2. First, check if direct substitution is possible by plugging in $x=2$:

1. **Problem Statement:** Given the function $$d(s) = s^3 + s - 4$$, find the number of local maxima of the function $$n(s)$$, where $$n(s)$$ is related to $$d(s)$$ (assuming $$n(s

1. Problem: Find the derivative $\frac{dy}{dx}$ of $y = 5x^4$. 2. Formula: Use the power rule for derivatives, which states that if $y = ax^n$, then $\frac{dy}{dx} = a n x^{n-1}$.