∫ calculus

1. **State the problem:** We have the function $$f(x) = 2x^3 - 12x^2 + 18x - 8$$ and need to find where it is increasing, decreasing, and its local extrema. 2. **Find the derivativ

1. **State the problem:** Find the absolute maximum and minimum values of the function $$f(t) = t - \sqrt[3]{t}$$ on the interval $$[-1,7]$$. 2. **Find the derivative:** To find cr

1. **State the problem:** Find the critical numbers of the function $$p(t) = t e^{5t}$$. Critical numbers occur where the derivative is zero or undefined. 2. **Find the derivative:

1. **State the problem:** Find the absolute maximum and absolute minimum values of the function $$f(x) = x^{-2} \ln(x)$$ on the interval $$\left[\frac{1}{2}, 5\right]$$. 2. **Find

1. **State the problem:** Find the limit $$\lim_{x \to 1} \frac{x^2 - 1}{x - 1}$$. 2. **Factor the numerator:** Recognize that $$x^2 - 1$$ is a difference of squares, so

1. The problem asks for the limit of the function $f(x) = \frac{1}{x}$ as $x$ approaches $0$ from the positive side, written as $\lim_{x \to 0^+} \frac{1}{x}$.\n\n2. When $x$ appro

1. **State the problem:** Find the limit $$\lim_{x \to \infty} \frac{2x + 1}{x + 3}$$. 2. **Analyze the expression:** As $$x$$ approaches infinity, the terms with the highest power

1. The problem asks for the limit of the function $$\frac{\sin x}{x}$$ as $$x$$ approaches 0. 2. This is a classic limit in calculus often used to define the derivative of the sine

1. **State the problem:** We are given the function $$f(x,y) = e^{2x} \cos y$$ and need to find the partial derivatives $$\frac{\partial f}{\partial x}$$ and $$\frac{\partial f}{\p

1. **State the problem:** We have a function $f(x,y) = x^2 y$ where $x = e^t$ and $y = \ln t$ for $t > 0$. We want to find $\frac{df}{dt}$ using the chain rule and express the answ

1. **State the problem:** We need to find the derivative $y'$ of the function $$y = (\sin x)^{\cos x}.$$\n\n2. **Rewrite the function using logarithms:** To differentiate a functio

1. **State the problem:** We are given the function $$w = x^2 y + y^2$$ where $$x = e^{5t}$$ and $$y = \sin t$$. We need to find $$\frac{dw}{dt}$$ using the chain rule and simplify

1. **Problem:** Find the tangent line to the function $f(x) = x^2 - 4x + 1$ at the point where $x=0$. Step 1: Calculate $f(0)$.

1. The problem is to find the integral of the function $x + 4$ with respect to $x$. 2. Recall that the integral of a sum is the sum of the integrals: $$\int (x + 4) \, dx = \int x

1. The problem is to find the integral of the function $x + 4$ with respect to $x$. 2. Recall that the integral of a sum is the sum of the integrals: $$\int (x + 4) \, dx = \int x

1. The problem is to evaluate the integral $\int x \, dx$. 2. Recall the power rule for integration: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ for $n \neq -1$.

1. The problem is to find the derivatives (Ableitung) of the two given functions: - $f(x) = 4x^3 - 2x + 1$

1. **Problem:** Show that the function $$f(x) = \frac{1}{x} \sin \frac{1}{x}$$ for $$x \neq 0$$ and $$f(0) = 0$$ is finite on $$[-1,1]$$ but not bounded, and determine any points o

1. **Problem statement:** We analyze the functions and limits given, solve equations, and study derivatives. 2. For $g(x) = x^3 + 3x + 8$, it has a unique root $\alpha$ in $(-2,0)$

1. The problem is to find the derivative of the function $f(x) = x^3$ with respect to $x$. 2. Recall the power rule for differentiation: if $f(x) = x^n$, then $f'(x) = nx^{n-1}$.

1. The problem appears to be finding the derivative of the function $f(x) = x^3$. 2. To find the derivative $\frac{d}{dx}(x^3)$, we use the power rule of differentiation which stat