∫ calculus

1. **State the problem:** Find the limit as $n$ approaches $+\infty$ of the expression $$\frac{3n^2 - n - 1}{(2n - 1)^2}.$$\n\n2. **Recall the formula and rules:** When evaluating

1. **Problem:** Determine if the series $$\sum_{n=1}^\infty \frac{1}{n^2}$$ converges or diverges using the Integral Test. 2. **Integral Test conditions:** The function $$f(x) = \f

1. **State the problem:** Find the absolute minimum value of the function $f(x) = \cos(x)$ on the interval $[0, \pi]$. 2. **Recall the function and interval:** The cosine function

1. **Stating the problem:** We have a function $f : \mathbb{R} \to \mathbb{R}$ defined by $$f(x) = \sqrt[3]{x^2}(3x - 7) = x^{\frac{2}{3}}(3x - 7).$$ We want to find values $a$ and

1. **State the problem:** We need to find the intervals where the function $f(x) = |9 - x^2|$ is decreasing. 2. **Understand the function:** The function is the absolute value of $

1. **State the problem:** The radius $r$ of a sphere is increasing at a rate of $\frac{dr}{dt} = 5$ cm/s. We need to find the rate at which the surface area $S$ is increasing when

1. **State the problem:** We need to find the first derivative $f'(x)$, the second derivative $f''(x)$, and then evaluate both at $x=1$ for the function $$f(x) = \frac{2x}{x^2 + 1}

1. **State the problem:** We are given the function $$f(x) = \frac{2x}{x^2 + 1}$$ and need to find its first derivative $$f'(x)$$, second derivative $$f''(x)$$, and then evaluate t

1. **State the problem:** Find the limit $$\lim_{x \to 4} \frac{x - 4}{x^2 - x - 12}$$. 2. **Recall the formula and rules:** When evaluating limits that result in an indeterminate

1. **State the problem:** Find the first-order partial derivatives $f_x$ and $f_y$ of the function $$f(x,y) = x^2 y + 3 x y^2 + 4.$$ 2. **Recall the formulas:**

1. **Problem statement:** Determine whether the integral $$\int_1^\infty \frac{2 + \sin x}{\sqrt{x}} \, dx$$ converges or diverges using the comparison theorem. 2. **Recall the com

1. **State the problem:** Evaluate the definite integral $$\int_0^3 \frac{x}{\sqrt{36 - x^2}} \, dx$$. 2. **Recall the formula and substitution:** This integral suggests a substitu

1. The problem is to find the first, second, and third derivatives of the function $y = \cos^3 x$. 2. Start with the first derivative. Use the chain rule: if $y = (u)^3$ where $u =

1. **Problem statement:** Evaluate the integral $$\int_0^\infty \frac{e^{at}}{(1+t^2)^2} \, dt$$ using Feynman's technique (differentiation under the integral sign). 2. **Recall Fe

1. **Stating the problem:** We want to evaluate the integral $$I = \int_0^\infty \frac{e^{ax}}{(1+t^2)^2} \, dt$$ where $a$ and $x$ are constants, using Feynman's technique (differ

1. The problem is to correctly identify the upper limit of the integral in part 8 (i), which is given as $\sqrt{2ax - x^2}$, not $2a \times \sqrt{2ax - x^2}$.\n\n2. This means the

1. **Problem statement:** Evaluate the double integral $$\int_0^{2a} \int_0^{2a\sqrt{2ax - x^2}} (x^2 + y^2) \, dy \, dx$$ by changing into polar coordinates. 2. **Recall polar coo

1. **Problem Statement:** Determine if the sequence $a_n = \frac{1}{n!}$ for $n \geq 1$ is convergent. 2. **Recall Definitions:**

1. **State the problem:** We need to test the convergence of the series with general term $a_n = \sqrt{n^2 + 1} - n$. 2. **Recall the convergence test:** A necessary condition for

1. The problem is to find the derivative of the function $$f(x) = x \arctan(\sqrt{x})$$ and verify if the given expressions for $$f'(x)$$ are correct and equivalent. 2. We use the

1. The problem asks for the limit of the function as $x$ approaches 0 based on the graph. 2. The limit of a function as $x$ approaches a value $a$ is the value that $f(x)$ approach