∫ calculus

1. **State the problem:** We want to find an approximate numerical value for the expression $$\frac{\theta \tan 2\theta}{1 - \cos 3\theta}$$ using small angle approximations, where

1. **State the problem:** We want to determine if a function is differentiable at $x=\frac{\pi}{2}$ using the definition of the derivative. 2. **Recall the definition of differenti

1. **State the problem:** We want to determine if the function $$f(x) = \frac{5\cos x}{2\tan x - 2}$$ is differentiable using the definition of the derivative: $$f'(x) = \lim_{h \t

1. **Stel het probleem vast:** We hebben een functie $$f(x) = \begin{cases} \frac{x^n - 1}{x - 1} & \text{als } x \neq 1 \\ a & \text{als } x = 1 \end{cases}$$

1. **State the problem:** Find all critical points of the function $$f(x) = x^3 - 6x^2 + 9x + a$$ where $$a \in \mathbb{R}$$. 2. **Recall the formula for critical points:** Critica

1. Planteamos el problema: Calcular el límite \(\lim_{x \to 0} \frac{x - \sin x}{x \sin x}\) (j) y \(\lim_{x \to 0} \frac{e^x - \sin x}{1 - \cos x}\) (f) usando la regla de L'Hôpit

1. **State the problem:** Find the possible values of $a$ such that the tangent to the curve $y = x^2 - 3$ at the point $(a,b)$ passes through the point $(0,-12)$.

1. **State the problem:** Approximate an interval for the sum of the convergent alternating series $$\sum_{n=1}^\infty (-1)^n \frac{2}{n^2}$$ using the Alternating Series Error Bou

1. **State the problem:** Determine if the infinite series $$\sum_{n=1}^\infty \frac{2^n - 1}{3^n}$$ converges. 2. **Rewrite the series:** Split the sum into two separate series:

1. The problem is to verify the integral of $x \ln x$ using integration by parts. 2. Recall the integration by parts formula: $$\int u\,dv = uv - \int v\,du$$

1. **State the problem:** We want to evaluate the integral $$\int \frac{\ln x}{\sqrt{x}} \, dx$$. 2. **Rewrite the integral:** Recall that $$\sqrt{x} = x^{1/2}$$, so the integral b

1. We start with the function $f(x) = (x^2 - 4)^2$. 2. To find the derivative $f'(x)$, we use the chain rule: if $f(x) = [g(x)]^2$, then $f'(x) = 2g(x) \cdot g'(x)$.

1. Het probleem is om te begrijpen hoe de afgeleide van de functie $f(x) = x e^{-3x}$ wordt berekend. 2. We gebruiken de productregel voor afgeleiden: als $f(x) = u(x) v(x)$, dan i

1. **State the problem:** We need to find the points where the given piecewise function is discontinuous. 2. **Recall the definition of discontinuity:** A function is discontinuous

1. The problem is to find the derivative of the composite function $f(x) = (u \circ s)(x) = u(s(x))$ where $$u(x) = x^2 + x^4$$

1. The problem is to find the derivative of the function $$f(x) = (3x + 2)^2 + (3x + 2)^4$$. 2. We use the chain rule for differentiation: if $$f(x) = g(h(x))$$, then $$f'(x) = g'(

1. **Problem 1a:** Find the area bounded by the curves $f(x) = 2x - x^2$ and $g(x) = x - 2$. 2. **Find the points of intersection:** Solve $2x - x^2 = x - 2$.

1. **Problem statement:** Find the area of the region bounded by the graphs of the functions $f(x) = 2x - x^2$ and $g(x) = x - 2$.

1. **State the problem:** Evaluate the integral $$\int_0^\pi \frac{e^x - 1}{e^x - x} \, dx.$$\n\n2. **Analyze the integrand:** Let $$f(x) = \frac{e^x - 1}{e^x - x}.$$ We want to fi

1. The problem asks to find the value of $x$ where the function $h(x)$ is continuous but not differentiable. 2. A function is continuous at $x=a$ if the left-hand limit, right-hand

1. **Problem statement:** Given the function $$y = x^4 - x^3 + 4x - 1,$$ find the second derivative $$\frac{d^2y}{dx^2}$$ and determine the values of $$x$$ for which $$\frac{d^2y}{