∫ calculus

1. **State the problem:** Find the derivative $f'(x)$ of the function $$f(x) = x^2 + 4x + \sqrt{x^3}.$$\n\n2. **Rewrite the function:** Recall that $$\sqrt{x^3} = x^{3/2}.$$ So, $$

1. **State the problem:** Differentiate the function $$f(x) = 4x^4 + 2x^3 + \sqrt{x} + 20$$ with respect to $$x$$. 2. **Recall the differentiation rules:**

1. **State the problem:** Find the area bounded by the curve $y = x^2$, the x-axis, and the vertical lines $x=1$ and $x=3$. 2. **Formula used:** The area under a curve $y=f(x)$ fro

1. The problem is to find the derivative of a function, which means determining the rate at which the function's value changes with respect to its input variable. 2. The derivative

1. **Problem statement:** Find the $n$-th derivative of the function $$y = x^{2m} e^{ax}$$ using Leibniz's rule. 2. **Recall Leibniz's rule for the $n$-th derivative of a product:*

1. The problem asks for the second-order Taylor expansion of a function $f(x,y)$ about the point $(a,b)$. 2. The general formula for the second-order Taylor expansion of a function

1. **Problem Statement:** Find the second-order Taylor expansion of the function $f(x,y)$ about the point $(a,b)$. 2. **Formula for the second-order Taylor expansion:**

1. Problem a) Find $$\int \frac{6}{x^2 + 3} \, dx$$ Formula: $$\int \frac{dx}{x^2 + a^2} = \frac{1}{a} \arctan \frac{x}{a} + C$$

1. Problem a) Find the integral $$\int \frac{6}{x^2 + 3} \, dx$$ Formula: $$\int \frac{dx}{x^2 + a^2} = \frac{1}{a} \arctan \frac{x}{a} + C$$

1. **State the problem:** We are asked to find the limits of the function $f(x)$ as $x$ approaches $-2$ and $2$ from the left and right, and the overall limit at these points, base

1. Masalah ini adalah menghitung integral dari fungsi $p$ terhadap $p$ dari 0 sampai 2, kemudian hasilnya dikalikan dengan $t$. 2. Integral yang diberikan adalah $$\int_0^2 p \, dp

1. Diketahui fungsi dua variabel $$A(t,p) = tp + t^3$$ dan domain integrasi $$0 \leq t \leq 3$$, $$0 \leq p \leq 2$$. 2. Kita diminta menghitung integral ganda:

1. We are asked to find the limits of polynomial, rational, and radical functions. 2. The limit of a polynomial function as $x$ approaches a value $a$ is simply the value of the po

1. **Problem Statement:** Find the absolute maximum value of the function $f(x)$ on the interval $[-4,6]$ given the derivative $f'(x)$ and some key points. 2. **Recall:** The absol

1. The problem is to understand the definite integral \(\int_a^b f(x)\,dx\), which represents the area under the curve of the function \(f(x)\) from \(x=a\) to \(x=b\). 2. The form

1. **Problem statement:** Find the derivative $\frac{dy}{dx}$ and the exact coordinates of point B. 2. **Step 1: Identify the function and point B.** Since the function or coordina

1. **State the problem:** We have the curve given by the function $$y=\frac{\cos x}{2-\sin x}$$ and need to find the coordinates of points A and C where the curve intersects the x-

1. **Problem Statement:** Prove that if there exists a number $M$ such that

1. The problem is to evaluate the improper integral $$\int_a^{+\infty} f(x) \, dx$$ which is defined as the limit $$\lim_{b \to \infty} \int_a^b f(x) \, dx$$. 2. The formula used i

1. **Problem:** Find the derivative of \(y = \frac{\sin x}{\cos x - \sin x}\). 2. **Formula:** Use the quotient rule for derivatives:

1. We are asked to find the limit $$\lim_{x \to 4} \frac{3 - \sqrt{5 + x}}{1 - \sqrt{5 - x}}.$$\n\n2. This is a limit involving square roots. Direct substitution gives \(x=4\): num