∫ calculus

1. We are given the derivative of a function: $f'(x)=\cos \left( \frac{x}{2}\right) -3\sin (x^2)$. We need to find the value of $x$ in the interval $(0,2)$ where the graph of $f$ h

1. The problem is to find the antiderivative (indefinite integral) of a given function, which means finding a function whose derivative is the original function. 2. The general for

1. The problem is to find the indefinite integral of the polynomial function $3x^2 + 2x - 1$ with respect to $x$. 2. The formula for integrating a power function $x^n$ is:

1. **Problem statement:** Calculate the double integral of the function $f(x,y) = 12xy^2 - 8x^3$ over the region $R = \{(x,y) : 1 \leq x \leq 2, -1 \leq y \leq 2\}$. 2. **Formula a

1. The problem is to solve an expression or equation using series expansion. 2. Series expansion involves expressing a function as an infinite sum of terms calculated from the valu

1. **State the problem:** Find the limit $$\lim_{x\to 2} \frac{\ln x - \ln 2}{x - 2}$$. 2. **Recall the formula:** This limit resembles the definition of the derivative of the func

1. **State the problem:** Find the derivative of the function $$f(x) = (x^2 + 1) \cdot \sin(x)$$ using the product rule. 2. **Recall the product rule formula:** If $$f(x) = u(x) \c

1. Masalah yang diberikan adalah mencari nilai limit dari fungsi $$\lim_{x \to -1} \frac{x^2 - 2x - 3}{x^2 - 3x - 4}$$. 2. Langkah pertama adalah substitusi langsung nilai $x = -1$

1. مسئله: بررسی وجود حد تابع $$f(x,y) = \frac{x^2 - y^2}{x^2 + y^2}$$ در نقطه $$(0,0)$$. 2. فرمول و نکات مهم: برای بررسی حد تابع دو متغیره در نقطه‌ای، باید حد تابع را از مسیرهای مخ

1. **State the problem:** We need to find $\frac{dy}{dx}$ for the implicit equation $$xy^2 + \cos y = e^{3x} - 4 \tan y.$$\n\n2. **Recall the rules:** We will use implicit differen

1. The problem states that the graph of the function $f(x) = 2x^3 - 3x^2 - 12x + 7$ is decreasing at the point $(-1, 2)$. We need to verify this by analyzing the derivative of the

1. **Problem statement:** Evaluate the limits:

1. The problem asks us to analyze the behavior of the function $f(x)$ near the vertical asymptotes at $x=0$ and $x=4$ based on the graph. 2. Vertical asymptotes occur where the fun

1. **State the problem:** Evaluate the definite integral $$\int_4^8 \left(6t^{\frac{5}{2}} - 3t^{\frac{3}{2}}\right) dt$$ using the Fundamental Theorem of Calculus, part 2. 2. **Re

1. **Problem Statement:** Given \(\lim_{x \to a} f(x) = 0\), \(\lim_{x \to a} h(x) = -7\), and \(\lim_{x \to a} g(x) = 3\), find the following limits if they exist. 2. **Recall Lim

1. **State the problem:** We have the curve $y = 9 - 4x - \frac{x}{2}$ for $x > 0$.

1. **Problem:** Find the derivative $y'$ of the implicit function defined by $$x^2 - xy + y^2 = 7.$$ 2. **Formula and rules:** We use implicit differentiation. Differentiate both s

1. The problem is to find the derivative $\frac{dy}{dx}$ of the implicit function given by the equation $$y^3 + y^2 - 5y - x^2 = -4.$$ 2. We use implicit differentiation. Different

1. The problem is to find the limit $$\lim_{x \to 0} \frac{\sin 3x}{x}$$. 2. We use the standard limit rule: $$\lim_{x \to 0} \frac{\sin x}{x} = 1$$.

1. We are asked to find the limit: $$\lim_{x \to 2^+} \frac{x + 2}{x^2 - 4}.$$ 2. The denominator can be factored using the difference of squares formula: $$x^2 - 4 = (x - 2)(x + 2

1. Evaluate the integral (a) $\int x^2 \cos 2x \, dx$. We use integration by parts: $\int u \, dv = uv - \int v \, du$.