∫ calculus

1. **State the problem:** Find the derivative $\frac{dy}{dx}$ of the function $$y = (2 + 6x)^{\frac{5}{x}}$$ and then evaluate the slope and the value of $y$ at $x=1$. 2. **Use log

1. **State the problem:** We are given the function $$y = \left(\frac{x}{7} + \frac{7}{x}\right)^7$$ and asked to:

1. **State the problem:** We have the function $$y = \frac{x^7 + 7x}{7}$$ and need to (a) decompose it into the form $$y = f(u)$$ and $$u = g(x)$$ with non-identity functions, and

1. **State the problem:** Find the total area of the shaded regions between the curves $y=2-x$ and $y=4-x^2$ over the interval where they intersect. 2. **Find the points of interse

1. **State the problem:** We analyze the given cubic-like function to find its relative maximum, relative minimum, intervals of increase and decrease, domain, and range. 2. **Ident

1. **State the problem:** We want to approximate the area under the curve of the function $f(x) = \sqrt{x^3 + 1}$ from $x=0$ to $x=2$ using the Midpoint (Mid-ordinate) Rule with si

1. **Problem Statement:** Find the values of $a$ and $b$ given that $$\lim_{x \to \infty} \sqrt{x^2 - x + 1 - 2x} = b$$ and $b$ is rational.

1. **State the problem:** We want to expand the function $$f(x) = (1+x)^{\frac{5}{2}}$$ using Maclaurin series up to the term in $$x^4$$ and then approximate $$f(0.2)$$. 2. **Recal

1. **State the problem:** We need to find the derivative of the product function $h(x) = f(x) \cdot g(x)$ at $x=1$, $x=2$, and $x=3$. The graph of $f(x)$ has a sharp corner at $x=2

1. The problem asks to find the limit $$\lim_{x \to 1} \frac{\sqrt[3]{x} - 1}{x - 1}$$. 2. This limit is of the form $$\frac{f(x) - f(a)}{x - a}$$ where $$f(x) = \sqrt[3]{x}$$ and

1. We are asked to find the limit $$\lim_{x \to 1} \frac{\sqrt[3]{x} - 1}{x - 1}$$. 2. This is a limit of the form $$\frac{f(x) - f(a)}{x - a}$$ where $$f(x) = \sqrt[3]{x}$$ and $$

1. **State the problem:** We want to find the Instantaneous Rate of Change (IROC) of the function $y = \frac{t^2}{4} - 2$ at $t = 15$ without using derivatives. 2. **Recall the for

1. **State the problem:** Find the domain of the vector function $$\mathbf{r}(t) = \langle \ln(t + 2), \frac{t}{\sqrt{36 - t^2}}, 2^t \rangle$$ in interval notation. 2. **Analyze e

1. **State the problem:** Find the limit as $t \to 0$ of the vector function $$\left\langle \frac{4e^t - 4}{t}, \frac{\sqrt{1+t} - 1}{t}, \frac{5}{1+t} \right\rangle$$

1. **State the problem:** We are given the function $$f(t) = 4 - 4t^2 + 4t^4 - 4t^6 + \cdots = \sum_{n=0}^\infty (-1)^n 4 t^{2n}$$ and the function $$F(x) = \int_0^x f(t) \, dt.$$

1. **Problem statement:** We analyze the variation of the function given its derivative:

1. **State the problem:** Differentiate the function $f(t) = 6e^t \left(5\sqrt{t} + \frac{1}{2t^7}\right)$ with respect to $t$. 2. **Recall the product rule:** If $f(t) = u(t)v(t)$

1. **State the problem:** Differentiate the function $$f(t) = 6e^t \left(5\sqrt{t} + \frac{1}{2t^7}\right)$$ with respect to $$t$$. 2. **Recall the product rule:** If $$f(t) = u(t)

1. The problem is to find the derivative of the function $f(x) = \ln(\sin x)$.\n\n2. We use the chain rule and the derivative of the natural logarithm function. The derivative of $

1. The problem is to evaluate the integral $$\int_0^{\frac{\pi}{4}} \tan^6 x \, dx$$. 2. We use the reduction formula for integrals of powers of tangent:

1. Problemet är att bestämma ett närmevärde för $h(10,1)$ givet att $h(10) = 3$ och $h'(10) = -2$. Vi ska använda derivatans definition för att lösa detta. 2. Derivatans definition