∫ calculus

1. **State the problem:** Evaluate the limit $$\lim_{x \to 8} \frac{\sqrt[3]{x-2} - 3}{x-8}$$. 2. **Recognize the form:** Substitute $x=8$ directly:

1. The problem is to find the limit of a function as $x$ approaches 0. 2. Since the user did not specify the function, let's consider a common example: $\lim_{x \to 0} \frac{\sin x

1. Problem statement: Analyze the function $f(x)=\frac{3x^2}{x^2-1}$ including domain, intercepts, asymptotes, critical points, and local extrema. 2. Domain: The denominator is zer

1. Problem 26: Find the velocity at time $t_0$ given the position function $s(t)$. Velocity is the derivative of position with respect to time: $v(t) = \frac{ds}{dt}$.

1. The problem is to find the derivative of the polynomial function given above. 2. Recall that the derivative of a polynomial term $ax^n$ is $a n x^{n-1}$.

1. **State the problems:** - Find $$\lim_{x \to 0} \frac{\cos x}{x}$$.

1. Let's start by understanding what integration is. Integration is the process of finding the integral of a function, which can be thought of as the area under the curve of that f

1. The Taylor series is a way to represent a function as an infinite sum of terms calculated from the derivatives of the function at a single point. 2. Suppose we have a function $

1.1 **Problem:** Show that $$\sum_{k=0}^{n} k^{3} = \frac{n^{4}}{4} + \frac{n^{3}}{2} + \frac{n^{2}}{4}$$

1. **State the problem:** We want to find the limit $$\lim_{x \to 3} \frac{\frac{1}{x} - \frac{1}{3}}{x - 3}$$

1. **State the problem:** Find the volume of the tetrahedron bounded by the coordinate planes $x=0$, $y=0$, $z=0$ and the plane $3x + 6y + 4z - 12 = 0$ using double integration. 2.

1. The problem appears to involve differentiating or integrating expressions involving $\sin x$ and $t$. However, the input is unclear and incomplete. 2. Assuming you want to diffe

1. The problem is to find the limits of the function $3t^2$ as $t$ approaches $\sin x$ and $1$. 2. First, evaluate the limit as $t \to \sin x$:

1. **State the problem:** We want to evaluate the integral $$\int \sin x \left(1 + 3t^2\right) dt$$ and then differentiate the result with respect to $x$. Then, we want to differen

1. Let's start by reviewing the basics of calculus, including limits, derivatives, and integrals. 2. Limits: Understand how to find the limit of a function as the input approaches

1. The "area under the curve" refers to the region between a graph of a function and the x-axis over a certain interval. 2. It is often calculated using definite integrals in calcu

1. **State the problem:** We need to find the total surface area of the solid formed by rotating the curve given by $$y=\sqrt{25-x^2}$$ about the x-axis for $$x$$ in the interval $

1. The problem is to determine the integral of $y \, dx$. 2. To solve this, we need the function $y$ expressed in terms of $x$. Without a specific function for $y$, the integral ca

1. The problem is to find the derivative $\frac{dy}{dx}$ of the function $$y = \frac{6}{7 + x^{2}}.$$\n\n2. Rewrite the function as $$y = 6(7 + x^{2})^{-1}$$ to apply the chain rul

1. The problem asks which statement guarantees the existence of a number $c$ in the interval $[-2, 3]$ such that $f(c) = 10$. 2. Statement A: $f$ is increasing on $[-2, 3]$ with $f

1. The problem asks to solve an equation or integral using the method of change of variable (substitution). 2. To apply this method, identify a substitution variable $u$ that simpl