∫ calculus

1. **State the problem:** Find the limit $$\lim_{x \to \frac{1}{3}} \left(9x^2 - \frac{1}{27}x^3 - 1\right).\n\n2. **Recall the limit rule:** If the function is continuous at the p

1. The problem is to find the limit as $x$ approaches $\frac{1}{3}$ of a function, but the function is not specified. 2. To solve a limit problem, we need the function expression $

1. The problem is to find the limit as $x$ approaches 3 for the expression $$\frac{9x^2 - 1}{27x^3 - 1}$$. 2. The formula for limits is to substitute the value of $x$ into the expr

1. **State the problem:** Find the limit \( \lim_{x \to 4} \frac{x-4}{4-\sqrt{x+12}} \). 2. **Identify the issue:** Direct substitution gives \( \frac{4-4}{4-\sqrt{4+12}} = \frac{0

1. Let's start by understanding the problem: Points of inflection are points on a curve where the concavity changes, which means the second derivative of the function changes sign.

1. The problem is to analyze the function $$y = (x^2 - 10x)^4$$ and understand its critical points, concavity, and points of inflection. 2. To find critical points, we first find t

1. **Problem Statement:** We analyze the function $$y = x^3 + \frac{3}{x}$$ to find its critical points, concavity, and points of inflection.

1. **Problem Statement:** We are given the function $$y = x \sqrt{4 - x^2}$$ and information about its critical points, concavity, and points of inflection.

1. **Problem Statement:** Find the critical points, concavity, and points of inflection for the function $$y = x \sqrt{4 - x^2}$$ and graph it. 2. **Step 1: Rewrite the function fo

1. The problem is to study the function $$f(x) = \frac{e^x}{|x-1|}$$. 2. We will analyze the domain, intercepts, behavior near critical points, and limits.

1. **Problem Statement:** We want to understand and write Fubini's Theorem for double integrals in an exam setting.

1. **State the problem:** We have a cylindrical tank with radius $r=2.5$ feet being drained at a volume rate of $\frac{dV}{dt} = -0.25$ ft$^3$/sec (negative because volume is decre

1. **State the problem:** We need to solve the integral $$\int \sin(x) e^x \, dx$$ using integration by parts. 2. **Recall the integration by parts formula:**

1. **Problem Statement:** Find the derivatives of the following functions: a) $y = \sin(\cos(\tan x))$

1. **Stating the problem:** We need to evaluate the integral $$\int \frac{2 \cos x - 3 \sin x}{6 \cos x + 4 \sin x} \, dx.$$ 2. **Formula and approach:** When integrating a functio

1. **Problem Statement:** Evaluate the integral $$\int \frac{2 \cos x - 3 \sin x}{6 \cos x + 4 \sin x} \, dx.$$\n\n2. **Formula and Approach:** When integrating a function of the f

1. **Stating the problem:** We need to evaluate the double integral $$12 - \int_0^{\frac{\pi}{6}} \int_{\sin x}^{\frac{1}{2}} x y^2 \, dy \, dx$$ and then change the order of integ

1. **State the problem:** We need to evaluate the double integral $$12 - \int_0^{\frac{\pi}{6}} \int_{\sin x}^{\frac{1}{2}} x y^2 \, dy \, dx$$.

1. **State the problem:** Evaluate the expression $$12 - \int_0^{\frac{\pi}{6}} \int_{\sin x}^{\frac{1}{2}} xy^2 \, dy \, dx$$. 2. **Understand the integral:** This is a double int

1. **State the problem:** Evaluate the integral $$\frac{1}{2} \int_0^{\frac{1}{2}} \frac{y^2}{5} (\sin^{-1} y)^2 \, dy$$. 2. **Rewrite the integral:** We can factor out constants:

1. **Problem Statement:** Analyze the function $$f(x) = x^3 + 6x^2 + 9x + 2$$ for its relative extrema, intervals of increase/decrease, concavity, and points of inflection. 2. **Fi