∫ calculus

1. **State the problem:** A particle moves on the hyperbola defined by the equation $$xy = 15$$ for time $$t \geq 0$$ seconds. At a certain instant, $$x = 3$$ and $$\frac{dx}{dt} =

1. **State the problem:** Given the equation $$xy + 7x - 8x^2 = 5$$, we need to find the derivative $y'$ by implicit differentiation and then solve for $y$ explicitly and different

1. **Problem statement:** Determine if the function $$f(x) = \begin{cases} x^2 - 2x + 1, & x \leq -2 \\ \left(\frac{1}{3}\right)^x, & -2 < x \leq 3 \\ \log_3 x, & x > 3 \end{cases}

1. **State the problem:** Evaluate the double integral

1. **State the problem:** We are asked to evaluate the integral $$\int (3x - 24) \, dx$$ and also consider the substitution $$x = 2y$$. 2. **Recall the integral formula:** The inte

1. **State the problem:** Find the limit as $x$ approaches 1 of the expression $$\frac{1 - x^{\frac{2}{3}}}{1 - x^{\frac{1}{3}}}.$$\n\n2. **Recall the formula and rules:** When eva

1. **State the problem:** Find the limit $$\lim_{x \to 0} \frac{1}{x} \left( \frac{1}{2} - \frac{1}{3x + 2} \right)$$. 2. **Rewrite the expression inside the parentheses:**

1. **State the problem:** Find the limit $$\lim_{x \to 0} \frac{\ln(x+1)^2}{2x}$$. 2. **Rewrite the expression:** Note that $$\ln(x+1)^2 = (\ln(x+1))^2$$.

1. **State the problem:** We need to find the left-hand limit of the function $f(x)$ as $x$ approaches $-2$, denoted as $\lim_{x \to -2^-} f(x)$. 2. **Understanding limits from gra

1. **Stating the problem:** We want to evaluate the double integral $$\int_0^\infty \int_0^{\frac{\pi}{2}} \frac{x \sin \theta \ln(1 + x^2 \cos^2 \theta)}{(1 + x^2 \sin^2 \theta)^{

1. **Stating the problem:** We need to find the derivative of the function $f(x) = 7\ln x + 2$. 2. **Recall the derivative rules:**

1. The problem is to find the second derivative $f''(x)$ of the function $f(x) = 7x^3 - 6x^5$. 2. Recall the power rule for derivatives: if $f(x) = x^n$, then $f'(x) = nx^{n-1}$.

1. **State the problem:** Differentiate the function $v(t) = V(1 - e^{-t/T})$ with respect to $t$. 2. **Recall the differentiation rules:**

1. The problem is to find the integral of $\cos 3x$ with respect to $x$, i.e., compute $\int \cos 3x \, dx$. 2. Recall the formula for integrating cosine of a linear function: $\in

1. **State the problem:** We need to find the indefinite integral $$\int (6e^{2x} + 6x) \, dx$$. 2. **Recall the integration rules:**

1. **State the problem:** Differentiate the function $$f(x) = \frac{(x^2 + 1)^3}{x}$$ with respect to $x$. 2. **Rewrite the function:** To differentiate more easily, express $f(x)$

1. The problem is to differentiate a function using the product rule. 2. The product rule states that if you have two functions $u(x)$ and $v(x)$, then the derivative of their prod

1. The original function is $v = \sqrt{4x^2 - 1}$. 2. You found the derivative $v' = 4x(4x^2 - 1)^{-1/2}$.

1. The problem is to find the derivative $v'$ of the function $v = \sqrt{4x^2 - 1}$. 2. We use the chain rule for derivatives: if $v = (f(x))^{1/2}$, then $v' = \frac{1}{2}(f(x))^{

1. **State the problem:** Differentiate the function $$y = e^{2x} \times \sqrt{4x^2 - 1}$$ using the chain rule. 2. **Recall the product rule and chain rule:**

1. Problem from Seite 51 Nummer 8: Given the function $f(x) = -x^3 + 3x^2$, we are to split the area enclosed by the graph of $f$ and the x-axis by a vertical line so that the two