∫ calculus

1. **State the problem:** Find all first and second partial derivatives of the function $$f(u,x,y) = e^{u + x y^2}$$. 2. **Recall the formula:** For a function $$f = e^g$$, the der

1. **State the problem:** Find the limit as $n$ approaches infinity of the expression $$\frac{n^3 + n}{n^2 + n - 1}.$$\n\n2. **Recall the rule for limits of rational functions:** W

1. **State the problem:** Find the limit as $n$ approaches infinity of the expression $$\frac{n^3 + 5}{n^4 + n - 1}.$$\n\n2. **Recall the rule for limits of rational functions:** W

1. **State the problem:** We need to evaluate the integral $$\int - \frac{2 \cdot (x^2 - 8x - 8)}{x \cdot (x+2) \cdot (3x + 2)} \, dx.$$

1. **State the problem:** Find the derivative $y'$ of the function $$y = (x+1)e^x \sin x.$$\n\n2. **Recall the product rule:** For functions $u(x)$ and $v(x)$, the derivative of th

1. **Problem statement:** Write the expression $$\frac{10x^2 + 41x + 37}{(x+1)(x+2)(x+3)}$$

1. **State the problem:** We need to find \( \frac{dy}{du} \) when \( y = \frac{u^2 + 1}{u^2 - 1} \). 2. **Formula used:** To differentiate a quotient \( \frac{f(u)}{g(u)} \), use

1. **Problem Statement:** We have a function $g$ defined on the interval $-11 \leq x \leq 7$ by

1. **State the problem:** Find the derivative $y'$ of the function $$y = (2 + e^{3x})(1 - e^{-x}).$$ 2. **Formula used:** We will use the product rule for derivatives, which states

1. The problem asks to evaluate the limit of a matrix as $x$ approaches $a$, specifically $$\lim_{x \to a} \begin{bmatrix} a & b \\ c & d \end{bmatrix}.$$\n\n2. Since the matrix en

1. **State the problem:** We are given the limit expression

1. **State the problem:** We want to approximate the area under the curve $$y = \frac{5}{2} \sqrt{4 - x^2}$$ on the interval $$0 \leq x \leq 2$$ using a left endpoint Riemann sum w

1. **State the problem:** We need to find the derivative $\frac{d}{dx}$ of the function $$y = \frac{(x^2 + x)(x + 1)}{(x^3 - 1)(x^4 + 1)}.$$\n\n2. **Rewrite the function:** Simplif

1. Find the derivative of $f(x) = x^3$ using the limit definition. Step 1: Write the limit definition of the derivative:

1. **State the problem:** Find the derivative $\frac{dy}{dx}$ of the function $$y = \frac{(x^2 + x)(x + 1)}{(x^3 - 1)x^4 + 1}.$$\n\n2. **Rewrite the function for clarity:** Let $$y

1. The problem is to evaluate or understand the integral $$\int_{Birth}^{Death} f(Life)\,dt$$ which represents the accumulation of some function $f(Life)$ over the time interval fr

1. **State the problem:** Find the derivative $\frac{dy}{dx}$ of the function $$y = \frac{2x^3}{(x+1)x^2}.$$\n\n2. **Simplify the function first:** The denominator is $(x+1)x^2 = x

1. **Problem statement:** A woman at point A on the shore of a circular lake with radius 2 miles wants to reach point C, diametrically opposite A, by first rowing to a point B on t

1. **State the problem:** Find the derivative $\frac{dy}{dx}$ where $y= \frac{6(x^2+1)}{(x+1)(x+2)}$. 2. **Formula and rules:** We will use the quotient rule for derivatives: if $y

1. Let's clarify the problem: You want to know which types of functions are generally easier to differentiate and which are easier to integrate. 2. Differentiation and integration

1. Let's start by stating the problem: Integration by parts is a technique used to integrate products of functions. 2. The formula for integration by parts is derived from the prod