∫ calculus

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

1. **Enunciado do problema:** Calcule o volume da região abaixo do gráfico da função $$f(x,y) = \begin{cases} \frac{x^2 - y^2}{\sqrt{x^2 + y^2}} & \text{se } (x,y) \neq (0,0) \\ 0

1. State the problem: Find $$\lim_{x\to 3}\frac{x^2-9}{x-3}.$$\n\n2. Substitute directly (quick check): The form is $$\frac{0}{0}$$ because $$x^2-9=(x-3)(x+3).$$\n\n3. Factor the n

1. **State the problem:** Find the limit $$\lim_{x \to -3} \frac{ \sqrt{2}\sqrt{x^2 - x} - \sqrt{6}\sqrt{x^2 + x - 6} }{ x^2 - 4x + 3 }$$

1. **Problem statement:** We need to find the least positive integer $m$ such that the temperature $P(m)$ of the ingot is below 60°C. 2. **Given function:**

1. **State the problem:** We have the function $f(x) = -\frac{1}{3}x^3 + 2x^2 - 12$.

1. **Problem statement:** Given the function $f(x) = -\frac{1}{3}x^3 + 2x^2 - 12$, find: i) The intervals where $f$ is increasing or decreasing and the local maxima and minima.

1. Problem: Evaluate $$\lim_{x\to 3}\,\frac{x^2-9}{x-3}.$$\n

1. State the problem: Evaluate the limit $\lim_{x\to 2} \frac{x^2-4}{x-2}$. 2. Identify the type of limit: Plugging in $x=2$ gives an indeterminate form $\frac{0}{0}$.

1. **Problem statement:** Given the function $f(x) = -\frac{1}{3}x^3 + 2x^2 - 12$, find: i) The intervals where $f$ is increasing or decreasing and the local maxima and minima.

1. **State the problem:** We need to find the derivative of the function $$f(x) = \frac{x}{x+1}$$ using the definition of the derivative. 2. **Recall the definition of the derivati

1. **State the problem:** We need to find the indefinite integral of the function $3x + e^x$ with respect to $x$. 2. **Formula used:** The integral of a sum is the sum of the integ

1. **State the problem:** Evaluate the integral $$\int 6 \cdot 2^{3x} \, dx$$. 2. **Recall the formula:** The integral of an exponential function $$a^{kx}$$ with respect to $$x$$ i

1. **State the problem:** We need to find the derivative of the function $$f(x) = \frac{x}{x+1}$$ using the definition of the derivative. 2. **Recall the definition of the derivati

1. **State the problem:** We need to find the derivative of the function $$f(x) = x^{x+1}$$ using the definition of the derivative. 2. **Recall the definition of the derivative:**

1. **State the problem:** We need to find the integral of the function $5e^{2x}$ with respect to $x$. 2. **Recall the formula:** The integral of $e^{ax}$ with respect to $x$ is giv

1. **State the problem:** Evaluate the integral $\int \cos(2x - 4x) \, dx$. 2. **Simplify the argument of cosine:**

1. **State the problem:** We are given the acceleration function of a particle as $a(t) = 36t^2 + 12t$. We need to find the velocity at time $t=2$ given that the velocity at $t=1$

1. The problem states that the derivative of a function $f$ is given by $f'(x) = x^4 + 3x^2 - 2$ and that $f(0) = 4$. We need to find the value of $f(1)$. 2. To find $f(1)$, we fir

1. **Problem statement:** Use the Intermediate Value Theorem (IVT) to verify that $f(x)$ has a zero in the given interval, then use the bisection method to find an interval of leng

1. **State the problem:** Given \( \frac{dy}{dx} = 7x^{2} - 5 \) and \( \frac{dx}{dz} = 6x - 4 \), find \( \frac{dy}{dz} \). 2. **Formula used:** By the chain rule for derivatives,