∫ calculus

1. **State the problem:** We need to find the limit $$\lim_{x\to 0} h(x)$$ based on the graph description of the function $h$. 2. **Recall the definition of a limit:** The limit $$

1. **State the problem:** We need to find the limit $$\lim_{x \to -3} (g(x) f(x))$$ where $$g$$ and $$f$$ are given piecewise functions graphed around $$x = -3$$. 2. **Understand t

1. Let's first state the problem: You want to identify whether a given differentiation rule is the chain rule, product rule, or quotient rule. 2. The **product rule** is used when

1. **Problem statement:** Find the limit $$\lim_{x \to 0} \frac{1-\cos 2x}{x}$$. 2. **Formula and rules:** Recall the Taylor expansion for cosine near zero: $$\cos y = 1 - \frac{y^

1. The three symbols that look like longer $s$'s are called integral signs, written as $$\int$$. 2. These integral signs represent integration in calculus, which is a way to find a

1. Let's start by understanding what calculus is about. Calculus is the study of how things change and it involves two main concepts: differentiation and integration. 2. Differenti

1. **State the problem:** We need to evaluate the definite integral $$\int_1^4 \sqrt{x} \, dx$$. 2. **Recall the formula:** The integral of $$\sqrt{x}$$, which is $$x^{1/2}$$, is g

1. **State the problem:** Evaluate the integral $$\int_0^\infty \frac{\sin^2(x) \cos(x)}{e^x} \, dx.$$\n\n2. **Rewrite the integral:** Note that $$e^x = \exp(x)$$ and the integral

1. **State the problem:** Differentiate the function $$y = -3\frac{x^2 - 2}{5x + 2}$$ using the quotient rule. 2. **Recall the quotient rule formula:** For a function $$y = \frac{u

1. **Problem:** Find the limit as $x$ approaches $0$ of $$\frac{e^{4x}-1}{x}$$. 2. **Use a standard limit formula:**

1. **State the problem:** We want to find the limit $$\lim_{x \to 0} \frac{e^{4x} - 1}{x}.$$\n\n2. **Recall the formula and rules:** The exponential function $e^u$ can be expanded

1. The problem is to evaluate the integral $$\int_{-\infty}^{\infty} e^{-x^2} \, dx$$. 2. This is a classic Gaussian integral. The formula for the Gaussian integral is:

1. **State the problem:** We have a rectangular box with height $b$, length $b+4$, and width $b+1$. We want to find how fast the total surface area increases as $b$ increases. 2. *

1. **State the problem:** We are given the integral $$\int_1^3 f(x) \, dx = 7$$ and need to evaluate $$\int_1^3 (4f(x) - 3) \, dx$$. 2. **Recall the properties of integrals:** The

1. **State the problem:** We need to evaluate the integral $$\int \frac{2x + 1}{(x - 1)(x + 3)} \, dx.$$ by decomposing the rational function into partial fractions first. 2. **Par

1. **State the problem:** We need to evaluate the integral $$\int x(x^2 + 2)^3 \, dx.$$ 2. **Choose a substitution:** Let $$u = x^2 + 2$$ because the expression inside the parenthe

1. **State the problem:** We need to evaluate the integral $$\int (4x - 2x^{-\frac{1}{3}} + \frac{5}{x^2}) \, dx.$$ 2. **Recall the anti-differentiation formulas:**

1. The problem is to find the derivative of the function $\cos x$ with respect to $x$. 2. The formula for the derivative of the cosine function is:

1. ปัญหาคือการหาค่าอนุพันธ์อันดับสอง $y''$ ของฟังก์ชัน $y = 2e^{x^2} + \sin 5x$. 2. สูตรที่ใช้คืออนุพันธ์ของฟังก์ชันประกอบและอนุพันธ์ของฟังก์ชันตรีโกณมิติ:

1. ปัญหาคือการหาค่าอนุพันธ์ $\frac{dy}{dx}$ เมื่อ $x=1$ โดยที่ $y$ เป็นฟังก์ชันของ $u$ และ $u$ เป็นฟังก์ชันของ $x$. 2. สูตรที่ใช้คือกฎลูกโซ่: $$\frac{dy}{dx} = \frac{dy}{du} \times

1. **โจทย์:** กำหนดให้ $f(x) = x^2 + 3x - 4$, $g(x) = 5 - 6x$, และ $h(x) = 7x + 8$