∫ calculus

1. **Problem Statement:** We are given the position function $s(t)$ of a car along a straight road and asked to identify which graph (A-F) best represents the derivative of the vel

1. **State the problem:** Find the limit $$\lim_{x \to \infty} \frac{2x^3 - 5x + 1}{4x^2 + 3x - 2}$$. 2. **Recall the rule for limits at infinity of rational functions:** When $x \

1. **State the problem:** Evaluate the first integral $$\int_0^2 \frac{1}{(2x - 1)^{3/2}} \, dx$$ since the other integrals are either improper or complex and the user asked to sol

1. **State the problem:** We need to evaluate the definite integral $$\int_1^{10} \frac{2624}{x^2 + x + 1} \, dx.$$\n\n2. **Recall the formula and approach:** The integral involves

1. The problem asks to find the limits of the function $f(x)$ as $x$ approaches several points where the function has discontinuities. 2. Recall that the limit $\lim_{x \to a} f(x)

1. **State the problem:** Find the derivative of the function $f(x) = 5x - x^2$ using the difference quotient method. 2. **Recall the difference quotient formula for the derivative

1. **Problem Statement:** Two roads A and B intersect perpendicularly. A diversion road C passes through a community located 4 km from road A and 6 km from road B. Find the smalles

1. **State the problem:** Determine which statement is true about the extreme points of the function $$f(x) = x^3 - 3x$$. 2. **Find the first derivative:** To find extreme points,

1. **State the problem:** Differentiate the function $f(x) = x e^{-x}$. 2. **Recall the formula:** To differentiate a product of two functions, use the product rule:

1. We are asked to find the derivative $\frac{dy}{dx}$ when $y = \cosh^{-1}(\sqrt{5x} + 5)$.\n\n2. Recall the formula for the derivative of the inverse hyperbolic cosine function:

1. The problem asks which method to use to find the volume of the solid formed by revolving the region bounded by $y=x^2$, $y=4x - x^2$, and $x=4$ about two different lines: $y=0$

1. The problem is to understand the notation and meaning of limits at infinity for a function $\phi(x)$. 2. When we write $$\lim_{x \to -\infty} \phi(x) = M,$$ it means that as $x$

1. **State the problem:** We are given a function $\varphi(x)$ graphed with behavior near $y=4$ at both ends and vertical asymptotes near $x=0$. We need to find the limits: (a) $\l

1. **State the problem:** We are asked to find the limits of the function $f(x)$ as $x$ approaches 4 from the left and right, the overall limit at $x=4$, and the value of $f(4)$ ba

1. **State the problem:** We need to find the limits of the function $f(x)$ as $x$ approaches $-2$, $0$ from the left and right, and $2$ from the left and right. 2. **Recall the de

1. **Problem Statement:** Given the function $u = x^2 + y^2 + z^2$ where $x = e^t$, $y = e^t \cos t$, and $z = e^t \sin t$, find the derivative $\frac{du}{dt}$. 2. **Formula Used:*

1. **State the problem:** Given the function $u = x^2 + y^2 + z^2$ where $x = e^t$, $y = e^t \cos t$, and $z = e^t \sin t$, find the derivative $\frac{du}{dt}$. 2. **Recall the cha

1. ปัญหาคือการวิเคราะห์บริเวณ R ที่ถูกปิดล้อมด้วยเส้นโค้งในพิกัดโพลา รที่กำหนดโดยสมการ $r = 5 - 5\cos\theta$. 2. สมการนี้เป็นสมการของรูปหัวใจ (cardioid) ในพิกัดโพลา โดย $r$ คือระยะ

1. **Problem Statement:** Evaluate the integral $$\int \frac{\ln x}{x} \, dx$$. 2. **Formula and Rules:** Recall the integral of the form $$\int \frac{\ln x}{x} \, dx$$ can be appr

1. **Problem:** Find the maximum and minimum values of the function $y = x^3 - 3x + 2$. 2. **Formula and rules:** To find maxima and minima, we use the first derivative test. Find

1. **Problem:** Find the maximum and minimum values of the function $$y = x^3 - 3x + 2$$. 2. **Formula and rules:** To find maxima and minima, we use the first derivative test. Fin