∫ calculus

1. **Problem statement:** Find the candidate points for extrema and verify the necessary and sufficient conditions for extrema of the function given its derivatives. 2. **Given:**

1. **State the problem:** Find the derivative $f'(x)$ of the function $$y = \ln \left( \frac{(x+1)^4 (x^3+2)}{x-1} \right)$$ using logarithmic differentiation. 2. **Recall the loga

1. **Problem a:** Find the equation of the tangent line to the curve defined by $$x^2 y + y^2 x = 6$$ at the point $$P(1,2)$$. 2. **Formula and rules:** To find the tangent line, w

1. **State the problem:** We need to find the derivative of the function $$f(x) = \sqrt{x} - 2$$. 2. **Recall the formula:** The derivative of $$x^n$$ with respect to $$x$$ is $$nx

1. The problem is to find the inflection points of a function, which are points where the concavity changes. 2. To find inflection points, we use the second derivative test: find w

1. **State the problem:** We need to find the value of the limit $$\lim_{x \to 4} \frac{f(x)}{x^2 - 16}$$ where the function $f(x)$ is given by a graph consisting of semicircles an

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 $$u(x) = x^2 + x^4$$