∫ calculus

1. The problem is to understand and use the notation \( \frac{dy}{dx} \), which represents the derivative of a function \( y \) with respect to \( x \).\n\n2. The derivative \( \fr

1. **State the problem:** We want to evaluate the limit $$\lim_{h \to 0} \frac{\frac{1}{x+h} - \frac{1}{x}}{h}.$$\n\n2. **Recall the formula:** This limit resembles the definition

1. **Problem Statement:** We are given the function $y = 2 \ln x$ and the area bounded by this curve, the line $y=2$, the $x$-axis ($y=0$), and the $y$-axis ($x=0$). We need to ske

1. Let's first clarify the problem: considering a horizontal strip in a mathematical context usually means analyzing a region bounded by two horizontal lines, say $y=a$ and $y=b$ w

1. Let's first understand what a horizontal strip means in a mathematical or graphical context. 2. A horizontal strip typically refers to a region bounded by two horizontal lines,

1. **State the problem:** Find the second derivative $y''$ of the function $y = e^{-x^2/2}$. 2. **Recall the formula:** To find $y''$, we first find the first derivative $y'$, then

1. **State the problem:** Find the derivative $y'$ of the function $$y = e^{-\frac{x^2}{2}}.$$\n\n2. **Recall the formula:** The derivative of $e^{u(x)}$ with respect to $x$ is $$\

1. **State the problem:** We have a curve defined by the function $$y = (x + 1)(2x - 3)^4$$ and want to find the approximate change in $$y$$ as $$x$$ increases from 2 to $$2 + p$$,

1. **Problem Statement:** We need to sketch the graphs of $y=2\ln x$ and $y=2x$, show the area bounded by these graphs, the x-axis, and the line $y=2$, and calculate the area and t

1. **Problem Statement:** We are given the graph of the function $y = 2 \ln x$ bounded by the line $y = 2$, the x-axis ($y=0$), and the y-axis ($x=0$). We need to sketch this graph

1. Problem: Find all first and second partial derivatives of $$f = \frac{xy}{x^2 + y^2}$$. Formula: Use the quotient rule for partial derivatives: $$\frac{\partial}{\partial x} \le

1. (a) Evaluate the indefinite integral $\int \left(2x^2 + \frac{2}{x^4}\right) dx$. Step 1: Rewrite the integral using negative exponents:

1. (a) Evaluate the integral $$\int \left(2x^2 + \frac{2}{x^4}\right) dx$$. Step 1: Rewrite the integral using negative exponents:

1. **Stating the problem:** We need to find the area of the region bounded by the curve $x = y^2$, the line $y - x = -1$, and the horizontal lines $y = 1$ and $y = 2$.

1. (a) Evaluate the indefinite integral $\int \left(2x^2 + \frac{2}{x^4}\right) dx$. Formula: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ for $n \neq -1$.

1. (a) Evaluate the integral $$\int \left(2x^2 + \frac{2}{x^4}\right) dx$$. Step 1: Rewrite the integral using negative exponents:

1. Problem statement. We are asked to evaluate the integrals from the quiz.

1. (a) Evaluate the integral $$\int \left(2x^2 + \frac{2}{x^4}\right) dx$$. Step 1: Rewrite the integral using negative exponents:

1. **State the problem:** Find the limit as $n \to \infty$ of the expression

1. **Problem Statement:** Find all first and second partial derivatives of the function $$f(x,y) = \frac{xy}{x^2 + y^2}$$. 2. **Recall the formulas:**

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 at infinity:** When dealin