∫ calculus

1. Problem 4.1: Find the derivative of the function $f(x) = (2x - 1)^2 \sin(2x)$. 2. Use the product rule for derivatives: if $f(x) = u(x)v(x)$, then $f'(x) = u'(x)v(x) + u(x)v'(x)

1. **Problem 4.1:** Find the derivative of the function $f(x) = (2x - 1)^2 \sin(2x)$. 2. **Formula and rules:** Use the product rule for derivatives:

1. Find the derivative of the function $f(x) = (2x - 1)^2 \sin(2x)$.\n\nStep 1: State the problem: Differentiate $f(x) = (2x - 1)^2 \sin(2x)$.\nStep 2: Use the product rule: If $f(

1. **Problem a:** Find $$\lim_{x \to 4} \frac{x^2 - 16}{x - 4}$$ 2. **Step 1:** Recognize that direct substitution $$x=4$$ gives $$\frac{16 - 16}{4 - 4} = \frac{0}{0}$$ which is in

1. **Problem:** Given the derivative $f'(x) = x^2 (x - 2)(x - 3)^2 (x - 4)$, find an open interval where $f(x)$ is decreasing. 2. **Recall:** A function $f(x)$ is decreasing where

1. **State the problem:** Find the area of the region bounded by the graph of the function $f(x) = x^3 + 1$, the x-axis, and the vertical lines $x = -2$ and $x = 0$. 2. **Formula a

1. The problem is to evaluate the integral $$I = \int \tan(\tan t) \cdot t \, dt$$. 2. This integral involves a composition of trigonometric functions and a polynomial term, which

1. The problem is to evaluate the integral $$I = \int \tan(\tan t) \cdot t \, dt.$$\n\n2. This integral is quite complex because it involves a composition of the tangent function i

1. **Problem:** Find the derivative $\frac{dy}{dx}$ for $y = x \cos 2x$ using the product rule. 2. **Formula:** The product rule states that if $y = u(x)v(x)$, then

1. **Problem Statement:** Determine if the following improper integrals converge or diverge: (i) $$\int_4^{\infty} 8^x \, dx$$

1. Problem statement: Find the limit as $x$ approaches $-5$ of the function. $$\lim_{x\to -5} \frac{\frac{1}{5}+\frac{1}{x}}{10+2x}$$

1. Problem: Compute the limit as $x \to -5$ of $\frac{\frac{1}{5} + \frac{1}{x}}{10} + 2x$. 2. Formula and rules: We use the limit laws: $\lim_{x\to a}(f(x)+g(x))=\lim_{x\to a}f(x)

1. **State the problem:** Find the limit as $x$ approaches $-5$ of the expression $$\frac{\frac{1}{5} + \frac{1}{x}}{10 + 2x}.$$\n\n2. **Rewrite the expression:** The expression ca

1. **State the problem:** Find the limit as $x$ approaches $-5$ of the expression $$\frac{\frac{1}{5} + \frac{1}{x}}{10} + 2x.$$\n\n2. **Rewrite the expression:** The expression ca

1. **State the problem:** Find the limit as $x$ approaches $-5$ of the expression $$\frac{\frac{1}{5} + \frac{1}{x}}{10} + 2x.$$\n\n2. **Rewrite the expression:** The expression ca

1. **State the problem:** We need to find the limit as $x$ approaches 5 of the expression $$\frac{\frac{1}{5+x}}{10+2x}.$$\n\n2. **Rewrite the expression:** The expression can be s

1. **State the problem:** Find the limit $$\lim_{x \to 5^+} \frac{\frac{1}{5} + \frac{1}{x}}{10 + 2x}$$. 2. **Recall the limit properties:** If the function is continuous at the po

1. The problem is to find the derivative $f'(x)$ of the function represented by the blue line graph described. 2. The graph starts near $(-5,4)$, goes down to a minimum near $(-2,-

1. **Problem statement:** Find the equation of the tangent line to the curve $y=\cos x$ at $x=1$. 2. **Recall the formula for the tangent line:**

1. **Problem statement:** Find the equation of the tangent line to the curve $y=\cos x$ at the point where $x=1$. 2. **Formula used:** The equation of the tangent line to a functio

1. Problem 7: Evaluate $$\iint_D y^2 \, dA$$ where $$D = \{(x,y) \mid -1 \le y \le 1, -y - 2 \le x \le y\}$$. 2. The integral is set up as an iterated integral with respect to $$x$