∫ calculus

1. **State the problem:** Evaluate the double integral $$\int_0^1 \int_0^2 \frac{2xy}{(1+x^2)(1+y^2)} \, dy \, dx$$ using the trapezoidal rule with step sizes $h = k = 0.25$.\n\n2.

1. **Problem:** Find the derivative of the function $f(x)=\ln x$. 2. **Recall the derivative rule:** The derivative of the natural logarithm function $\ln x$ is given by

1. We are asked to evaluate the double integral $$\int_0^1 \int_0^2 \frac{2xy}{(1+x^2)(1+y^2)} \, dy \, dx$$ using the trapezoidal rule with step sizes $h = k = 0.25$. 2. First, id

1. State the problem: Evaluate the double integral $$ \int_0^1 \int_1^2 \frac{2xy}{(1+x^2)(1+y^2)} \, dy \, dx $$

1. We need to find the derivative of $y = e^{\sin x}$ with respect to $z = \cos x$ at $x = \frac{\pi}{2}$. 2. Using the chain rule, $$\frac{dy}{dz} = \frac{dy/dx}{dz/dx}.$$

1. We are given a function $$f(x) = 5x + 2$$ and we need to find the value of $$g'(12)$$ where $$g(x)$$ is the inverse function of $$f(x)$$. 2. Recall that if $$g$$ is the inverse

1. **State the problem:** We are given two functions: $f(x) = e^x$ and $g(x) = x^2$. We need to find the value of the derivative of $f$ evaluated at $g(2)$, i.e., find $f'(g(2))$.

1. **State the problem:** We are given two functions $$f(x) = \sqrt{x}$$ and $$g(x) = x^2 + 7$$ and asked to find the derivative of the composition $$(f \circ g)'(3)$$. 2. **Recall

1. Stating the problem: Given $$y = \cot(ax)$$ and the differential equation $$\frac{dy}{dx} + 4(1 + y^2) = 0$$, find the value of $$a$$. 2. Calculate $$\frac{dy}{dx}$$ from $$y =

1. The problem shows a matrix $J$ defined with trigonometric expressions: $$J = \begin{pmatrix}-2 \times 4 \times \sin(4) & \cos(4) - 4 \times \sin(4) \\ 2 \times \cos(2) + 1 & 1 \

1. **State the problem:** Given the derivative expressions y' = 6y - x^2

1. **State the problem:** We want to evaluate or simplify the integral $$\int \frac{\cos^n x - \delta_i^n x}{\sqrt{1 + \cos^n x}} \, dn$$

1. The problem is to find the interval where the function $$y=\frac{1}{16}-x^{2}$$ is decreasing. 2. The first step is to find the derivative of the function to analyze its increas

1. Let's clarify the problem you are asking about involving $-\infty$. 2. If the question relates to a limit or boundary behavior such as $\lim_{x \to a} f(x) = -\infty$, it means

1. The problem is to find the interval where the function $$y = \frac{1}{16} - x^2$$ is decreasing. 2. To determine where the function is decreasing, we first find its derivative w

1. **Problem statement:** Evaluate the double integral $$\int_0^1 \int_x^1 y^2 e^{xy} \, dy \, dx$$ with the order of integration as given (do $y$ first). 2. **Inner integral (with

1. The problem is to evaluate the double integral $$\int_0^1 \int_x^1 y^2 e^{xy} \, dy \, dx.$$\n\n2. We first focus on the inner integral with respect to $y$: $$\int_x^1 y^2 e^{xy

1. **Stating the problem:** We have a sequence defined by the partial sums $$a_n = \frac{1}{3+1} + \frac{1}{3^2+1} + \cdots + \frac{1}{3^n+1}$$ for $n=1,2,\ldots$. We want to deter

1. Problem (b): Evaluate \(\int \frac{3}{\sin \theta - 3 \cos \theta - 1} \, d\theta\) using the substitution \(t = \tan(\frac{\theta}{2})\), and then find \(\int \frac{7 \sin \the

### Problem 31 Find

1. **State the problem:** Evaluate the limit $$\lim_{x\to 2} \frac{\sqrt{x-2}}{x-2}$$. 2. **Analyze the expression:** When plugging in $x=2$, the denominator and numerator both bec