∫ calculus

1. **State the problem:** Prove the inequality $$0 < \frac{1}{x} \log \frac{e^x - 1}{x} < 1$$ for $$x > 0$$ using Lagrange's Mean Value Theorem (MVT). 2. **Recall Lagrange's MVT:**

1. **State the problem:** Find the limit $$\lim_{x\to 2}\frac{x^2-4}{x^2+9x+14}$$. 2. **Recall the formula and rules:** To find limits involving rational functions, first try direc

1. **State the problem:** Find the limit $$\lim_{x\to 5}\frac{x^2-25}{x^2-5x}$$. 2. **Recall the formula and rules:** When direct substitution results in an indeterminate form like

1. Stating the problem: We want to find the limit $$\lim_{x \to 0} \frac{\sqrt{x} + \sin x}{\ln x}$$. 2. Important note: The natural logarithm function $\ln x$ is only defined for

1. Stating the problem: We want to find the limit $$\lim_{x \to 0} \frac{\sqrt{x} - \sin x}{\ln x}$$. 2. Important note: The limit involves $x \to 0$, but $\ln x$ is only defined f

1. **State the problem:** Find the limit $$\lim_{x \to 0} \left( \sqrt{3x + 5} - \sqrt{3x - 3} \right).$$ 2. **Recall the formula and technique:** When dealing with limits involvin

1. **State the problem:** We need to analyze the function $$f(x) = \frac{1}{(x+1)^2} + 3$$ at $$x = -1$$ and determine the type of discontinuity. 2. **Recall the definition:** An i

1. **State the problem:** Determine the convergence of the series $$\sum_{n=2}^\infty \frac{\sin(n+1) + \cos n}{n^2 - n}$$. 2. **Rewrite the general term:** The term is $$a_n = \fr

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

1. **State the problem:** Find the derivative of the function $y = x(2 - e^x)^3$. 2. **Formula used:** We will use the product rule and the chain rule.

1. **State the problem:** Find the derivative of the function $$y = (1 - x)^4 (1 + x + x^2)^4$$. 2. **Formula used:** We will use the product rule for differentiation: $$\frac{d}{d

1. **State the problem:** Given the function $$x = x^3 + 3x^2 - 9x + 6$$, we need to find the first derivative, determine critical points, identify intervals of increase/decrease a

1. **State the problem:** Find the shaded area between the curve $y = x^2$ and the line $y = 2x + 3$ bounded by their points of intersection. 2. **Find points of intersection:** Se

1. The problem is to understand and use the formula for the slope of the tangent line to a curve at a point, given by the limit: $$m = \lim_{x \to x_0} \frac{y - y_0}{x - x_0}$$

1. **Problem statement:** Find the equation of the tangent line to the function $$f(x) = \sqrt{25 - x^2}$$ at the point where $$x = 4$$. 2. **Formula and rules:** The slope of the

1. The problem is to find the derivative of the function $$f(x) = \frac{3x^2 + 2x - 1}{x - 4}$$. 2. We use the quotient rule for derivatives, which states:

1. Masalah: Diberikan fungsi $f(x) = 6x^5 + 3x^3 - 2x - 8$, kita diminta mencari turunan fungsi tersebut. 2. Rumus turunan: Untuk fungsi polinomial $f(x) = ax^n$, turunan adalah $f

1. **State the problem:** We need to evaluate the integral $$L=\int_0^{1.3} \sqrt{1+0.093636x^{-0.98}}\,dx.$$\n\n2. **Understand the integral:** The integrand is $$\sqrt{1+0.093636

1. **State the problem:** We need to evaluate the definite integral $$L=\int_0^{1.3} \left(1+0.093636x\right)^{-0.98} \, dx.$$\n\n2. **Recall the formula:** For an integral of the

1. **State the problem:** Find the arc length of the function $f(x) = 0.6x^{0.51} + 3$ over a given interval $[a,b]$. 2. **Formula for arc length:** The arc length $L$ of a curve $

1. **State the problem:** Find the arc length of the function $f(x) = 0.6x^{0.51} + 3$ over a given interval (assume from $x=a$ to $x=b$). 2. **Formula for arc length:** The arc le