∫ calculus

1. **Statement of the problem:** We study the function $g(x) = 2x - 1 - \ln x$ defined on $]0; +\infty[$ and analyze its behavior, zeros, and relation to $f(x) = x^2 - x\ln x$. The

1. **Problem:** Use the limit definition of the derivative to find $f'(x)$ if $f(x) = 3x + 1$. 2. **Limit definition of derivative:**

1. **State the problem:** Find the derivative of the function $$y=\frac{2x^3 + x}{3x - 1}$$. 2. **Recall the formula:** For a function $$y=\frac{u}{v}$$, the derivative is given by

1. Let's clarify the problem: You want to understand why the velocity function $v(t)$ is not considered explicitly simple. 2. In calculus and physics, a function is "explicitly sim

1. **Problem statement:** We are given the velocity function $v(t) = 2 \sin\left(e^{t/4}\right) + 1$ and acceleration $a(t) = \frac{1}{2} e^{t/4} \cos\left(e^{t/4}\right)$ for a pa

1. **State the problem:** Find the volume of the solid bounded by the sphere $$x^2 + y^2 + z^2 = 4a^2$$ and the cylinder $$x^2 + y^2 - 2ay = 0$$. 2. **Rewrite the cylinder equation

1. **State the problem:** Evaluate the limit $$\lim_{n \to \infty} \ln \left( \sqrt{9n^2 + 18n} - 3n \right).$$ 2. **Rewrite the expression inside the logarithm:**

1. The problem asks to evaluate the definite integrals \(\int_{-1}^2 f(x)\,dx\), \(\int_2^4 f(x)\,dx\), and \(\int_{-1}^7 f(x)\,dx\) by interpreting them as areas under the curve o

1. **Problem Statement:** Calculate the definite integrals of the piecewise function $f(x)$ over the intervals $[-1,2]$, $[2,4]$, and $[-1,7]$ based on the graph description.

1. Soalan a: Cari \(\frac{dy}{dx}\) bagi fungsi \(y = (6x^2 + 4x)^4\) dan nilai \(y\) bila \(x = \frac{1}{2}\). 2. Gunakan kaedah rantai untuk membeza fungsi kuasa komposit. Formul

1. The problem is to find the limit $$\lim_{x \to -\infty} \tanh(x) = \lim_{x \to -\infty} \frac{e^x - e^{-x}}{e^x + e^{-x}}$$ and understand why the sandwich theorem was not used.

1. **Problem Statement:** We need to find the area of the region enclosed by the given curves. Since the curves are not explicitly provided, let's assume the problem involves two f

1. **State the problem:** We need to find the area of the region enclosed by the curves $y = \cos x$ and $y = 2 - 2\cos x$ over the interval $0 < x < 2$.

1. **State the problem:** We need to find the area of the region enclosed by the curves $y=\sec^2 x$ and $y=8\cos x$ over the interval $-\frac{\pi}{3} \le x \le \frac{\pi}{3}$.

1. **Problem statement:** Find the limit $$\lim_{x \to 3} f(x)$$ where $$f(x) = \begin{cases} x - 1, & x \leq 3 \\ 3x - 7, & x > 3 \end{cases}$$ 2. **Formula and rules:** For piece

1. **Problem Statement:** Estimate the following limits using a table of values or direct substitution where possible. 2. **Recall:** The limit of a function $f(x)$ as $x$ approach

1. **Problem Statement:** Estimate or evaluate the given limits using tables of values or algebraic simplification. ---

1. **Problem Statement:** Estimate the limits using tables of values and evaluate given limits. 2. **Important formulas and rules:**

1. **State the problem:** We want to estimate the limits:

1. The problem asks whether the term "grow" in the phrase "f(t) must not grow faster than a function of exponential type" refers to convergence in the context of the Laplace transf

1. **State the problem:** Find the limit of the function $$f(x) = \frac{x^2 - 3x}{x^2 - 9}$$ as $$x$$ approaches 3 and -3 by evaluating the function at values close to these points