∫ calculus

1. **Problem:** Find the limit $$\lim_{n \to \infty} \frac{(3 - n)^4 - (2 - n)^4}{(1 + n)^4 - (n - 1)^4}$$

1. Find the derivative of $y = \frac{\sin x}{\cos x}$. Use the quotient rule: $$\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v u' - u v'}{v^2}$$ where $u = \sin x$ and $v = \cos x$

1. **State the problem:** Differentiate the function $f(x) = x - e^x$ with respect to $x$. 2. **Recall the differentiation rules:**

1. **State the problem:** Find the limit $$\lim_{x \to 1} \frac{\sin(x-1)}{x-1}$$. 2. **Recall the important limit formula:** The standard limit $$\lim_{t \to 0} \frac{\sin t}{t} =

1. **Stating the problem:** We have a function $f$ defined on the interval $[a,b]$ such that $f(x) \leq 0$ for all $x \in [a,b]$. We want to determine which of the given functions

1. **Problem Statement:** We need to identify which graph represents a continuous function $f$ such that $f(0) = 3$, $f'(2) = f'(-2) = 0$, and $f''(x) > 0$ for $-2 < x < 2$. 2. **U

1. **Problem Statement:** We are given a function defined on the interval $[0,2[$ and asked to determine whether it has an absolute minimum and/or maximum value on this interval.

1. **Problem Statement:** We analyze the function $f$ defined on the interval $[1,5]$ with the given curve characteristics. 2. **Given Information:**

1. **Problem:** Find the derivative of $y = \sin(3x) \cos(2x)$.\n\n2. **Formula:** Use the product rule: if $y = u v$, then $y' = u' v + u v'$. Here, $u = \sin(3x)$ and $v = \cos(2

1. The problem states that the given graph is the first derivative $f'(x)$ of a function $f(x)$, and we need to determine which of the provided graphs could represent the original

1. **Problem Statement:** Given the graph of the derivative function $f'(x)$ of a continuous function $f$ on $\mathbb{R}$, determine which statement about $f$ is true. 2. **Underst

1. **Problem Statement:** We are given the graphs of the derivatives $f'(x)$ and $g'(x)$ and asked to determine which graph could represent the function $h(x) = f(x) - g(x)$. 2. **

1. **State the problem:** Find the derivative of the function $y = \ln(4x + 3)$.\n\n2. **Recall the formula:** The derivative of $\ln(u)$ with respect to $x$ is given by $\frac{d}{

1. **State the problem:** Find the derivative of the function $y = \ln(4x + 3)$.\n\n2. **Recall the formula:** The derivative of $\ln(u)$ with respect to $x$ is given by $\frac{d}{

1. **State the problem:** Evaluate the integral $$\int \frac{1}{x} \, dx$$. 2. **Recall the formula:** The integral of $$\frac{1}{x}$$ with respect to $$x$$ is given by $$\int \fra

1. Find the derivative of the function (a) $g(x) = \int_2^x t^2 \sin t \, dt$. Step 1: State the problem.

1. **Problem:** Find the derivative of the function \( g(x) = \int_2^x x^2 \sin x \, dx \). 2. **Formula and rule:** By the Fundamental Theorem of Calculus, if \( F(x) = \int_a^x f

1. **Problem 1a:** Evaluate the integral $$\int \sqrt[3]{1 - x^2} \, dx$$ - This is an integral involving a cube root of a quadratic expression.

1. We are asked to evaluate the integral $$\int \frac{e^{\sqrt{x}}}{\sqrt{x}} \, dx$$. 2. To solve this, use the substitution method. Let $$t = \sqrt{x}$$, so $$x = t^2$$ and $$dx

1. The problem is to find the derivative of the function $y = e^{ax}$ where $a$ is a constant. 2. According to the derivative formula for exponential functions, the derivative of $

1. The problem asks to find the derivative of $\tan(ax)$ with respect to $x$. 2. According to the derivative formulas given, the derivative of $\tan(ax)$ is: