∫ calculus

1. **Problem:** Is the function $f(x) = \frac{4}{x - 3}$ continuous at $x = 3$? 2. **Step 1:** Recall the definition of continuity at a point $x = a$: A function $f$ is continuous

1. **State the problem:** We want to evaluate the definite integral $$\int_{-3}^{-1} (x^2 - 4x) \, dx$$ using the definition of the integral as a limit of Riemann sums. 2. **Recall

1. **State the problem:** Find the first derivative of the function $$y = x \arcsin(x) + \sqrt{1 - x^2}$$. 2. **Recall the formulas and rules:**

1. The problem is to find the area under a curve by integration. 2. The area under a curve $y=f(x)$ from $x=a$ to $x=b$ is given by the definite integral $$\text{Area} = \int_a^b f

1. **State the problem:** Find the first derivative of the function $$y = e^x \arctan(e^x)$$. 2. **Formula used:** We will use the product rule for derivatives, which states:

1. **Problem statement:** Find the value of $t$ where line $WY$ is normal to the curve $y=\frac{1}{2}x^{2}+1$ at point $B(2,4)$, then find the area of the shaded region bounded by

1. **State the problem:** Find the first derivative of the function $$y = \arcsin\left(\frac{a}{x}\right)$$ where $$x > 2$$. 2. **Recall the formula:** The derivative of $$y = \arc

1. **State the problem:** We want to find all solutions to the equation $$\sin(x) = x - 1$$ using Newton's method, accurate to six decimal places. 2. **Rewrite the equation:** Defi

1. **State the problem:** Find the limit $$\lim_{x \to \infty} \frac{\sqrt{x} - 3 - \sqrt{5x} + 1}{\sqrt{5x} - 1 - \sqrt{x} + 3}$$. 2. **Rewrite the expression:** Group terms to se

1. **State the problem:** Find the limit $$\lim_{x \to \infty} \frac{\sqrt{x} - 3 - \sqrt{5x} + 1}{\sqrt{5x} - 1 - \sqrt{x} + 3}$$. 2. **Rewrite the expression:** Group terms to se

1. **State the problem:** Evaluate the integral $$\int (4x^3 - 7x + 2) \, dx$$. 2. **Recall the formula:** The integral of a polynomial term $$ax^n$$ is given by $$\int ax^n \, dx

1. **State the problem:** Find the derivative of the function $$f(x) = \frac{4 - 5x}{2x^2 - 4x} \cdot (-x^4 + 3x - 2)(x^5 - 2x^4)$$

1. **Problem:** Find the first derivative of $$f(x) = (x^5 - 2x + 4)(3x^2 - 4x - 6)$$ 2. **Formula:** Use the product rule for derivatives: $$\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u

1. **Problem:** State Cauchy’s Mean Value Theorem. 2. **Statement:** Cauchy’s Mean Value Theorem states that if functions $f$ and $g$ are continuous on the closed interval $[a,b]$

1. **State the problem:** We want to find the area of the region $R$ enclosed by the curves $y = -x + 1$, $y = e^x$, $y = 0$, and the vertical line $x = 2$. 2. **Understand the reg

1. Evaluate the double integral $$\int_0^1 \int_y^1 e^{-x^2} \, dx \, dy$$. - The integral is over the region where $y$ goes from 0 to 1 and $x$ goes from $y$ to 1.

1. **State the problem:** Find the area of the region $R$ enclosed by the curves $y = -x + 1$, $y = e^x$, $y = 0$, and the vertical line $x = 2$ using a double integral. 2. **Under

1. **State the problem:** Find the area of the region $R$ enclosed by the curves $y = -x + 1$, $y = e^x$, $y = 0$, and the vertical line $x = 2$ using a double integral. 2. **Under

1. **State the problem:** Find the limit $$\lim_{x \to 0} \frac{x + a - a}{x}$$ where $a$ is a constant. 2. **Simplify the expression:** Notice that $a - a = 0$, so the expression

1. **Problem:** Find the limit $$\lim_{x \to 0} \frac{\sqrt{x + a} - \sqrt{a}}{x}$$ where $a > 0$. 2. **Formula and rule:** To evaluate limits involving square roots, multiply nume

1. **Problem Statement:** Evaluate the definite integral $$\int_0^1 \frac{x e^x}{(x+1)^2} \, dx$$. 2. **Method:** Use substitution to simplify the integral. Let $$u = x + 1$$, so $