∫ calculus

1. **Stating the problem:** We want to analyze the function $$h(x) = \arctan(\sin(\frac{1}{x^2}))$$ and understand its behavior, especially near the y-axis where $x$ approaches 0.

1. **Problem Statement:** Given the function $f(x) = x^2 \ln x$ for $x > 0$, we need to find:

1. **State the problem:** We need to check the validity of the Mean Value Theorem (MVT) for the function $f(x) = x^2 - 3x - 1$ on the interval $\left[-\frac{11}{7}, \frac{13}{7}\ri

1. **Problem Statement:** Given the function $f(x) = x^2 e^x$, we need to find: a) Intervals where $f$ is concave up and concave down.

1. **Problem:** Under what condition does the limit $\lim_{x \to a} f(x)$ exist and equal $L$? 2. **Formula and rule:** The limit $\lim_{x \to a} f(x) = L$ exists if and only if th

1. **Problem Statement:** Determine the correct answers for the limit problems given, including conditions for limit existence, evaluating limits from tables and graphs, and calcul

1. **Problem:** Find $\frac{dy}{dx}$ by implicit differentiation for the equation $$x^2 + y^2 = 100$$. 2. **Formula and rules:** When differentiating implicitly, treat $y$ as a fun

1. **State the problem:** We need to evaluate the improper integral $$\int_a^{+\infty} \frac{1}{15} e^{-\frac{x}{15}} \, dx$$ where $a$ is a constant.

1. **State the problem:** Find the derivative of the function $$y = \log_{10} \left( \frac{1+x}{1-x} \right)$$. 2. **Recall the formula:** The derivative of $$\log_a u$$ with respe

1. **State the problem:** Given the implicit equation $$x + xy - 2x^3 = 2$$, we need to find \(\frac{dy}{dx}\) by implicit differentiation (part a), solve for \(y\) as a function o

1. **State the problem:** We are given the function $y = x^3 \ln \sqrt{x^2 + 1}$ and need to find its derivative $\frac{dy}{dx}$. 2. **Rewrite the function:** Recall that $\sqrt{x^

1. **Problem:** Determine if the function $f(x) = x^4 + 3x^2 - 6x + 2$ is continuous at $x=3$. 2. **Formula and rules:** Polynomials are continuous everywhere. To check continuity

1. **State the problem:** Find the derivative with respect to $x$ of the function $$f(x) = x^3 \ln\sqrt{x^2 + 1}.$$\n\n2. **Rewrite the function for clarity:** Note that $$\sqrt{x^

1. The problem involves understanding the expression \(\sqrt{x}789\int_a^b f(x)\,dx\). 2. Here, \(\sqrt{x}\) means the square root of \(x\), which is \(x^{1/2}\).

1. Let's start by stating the problem: We want to rewrite and simplify the integral expression involving \( \frac{\sqrt{t^2-1}}{t} - t^2 \). 2. The expression is \( \frac{\sqrt{t^2

1. **State the problem:** We need to evaluate the integral $$\int \frac{x^2}{(x^2+1)(1+\sqrt{1+x^2})} \, dx.$$\n\n2. **Identify substitution:** Let us set $$t = \sqrt{1+x^2}.$$ The

1. **State the problem:** Find the extreme values of the function $f(x) = x^3 - 18x^2 + 96$ using the second derivative test. 2. **Recall the formulas and rules:**

1. **State the problem:** Find the extreme values of the function $f(x) = x^3 - 18x^2 + 96$ using the second derivative test. 2. **Find the first derivative:**

1. The problem asks: What does it mean for a sequence $x_n$ to be divergent? 2. In mathematics, a sequence $x_n$ is said to be divergent if it does not converge to a finite limit a

1. **Problem statement:** Given the implicit equation $$x + y + z = \log z,$$ find the partial derivatives $$\frac{\partial z}{\partial x}$$ and $$\frac{\partial z}{\partial y}.$$

1. **State the problem:** We are asked to find various limits and function values for a piecewise function based on the graph's behavior at specific points. 2. **Recall limit and f