∫ calculus

1. **State the problem:** Find the points of intersection of the line $y=\frac{1}{2}x$ and the parabola $y=2x-\frac{1}{6}x^2$. 2. **Set the equations equal to find intersection poi

1. The problem is to find the limit $$\lim_{x \to 0} \frac{\sin x}{x}$$. 2. We use the standard limit rule: $$\lim_{x \to 0} \frac{\sin x}{x} = 1$$.

1. **Problem statement:** (a)(i) Given the sequence $x_n = \frac{4n^2 + 3n + 99.5}{2n^2 + 5}$, find the limit $\ell$ as $n \to \infty$ assuming the sequence converges.

1. **State the problem:** Find all values of $x$ in the interval $0 < x < 2\pi$ for the function $f(x) = x - 2 \cos x$ where the slope of the tangent line is 2. 2. **Recall the for

1. **State the problem:** Find the limit $$\lim_{x \to 0} \frac{x^3 - 3x^2 + x}{x^3 - 2x}$$. 2. **Recall the limit rules:** When direct substitution results in an indeterminate for

1. **State the problem:** Find the limit $$\lim_{x \to 0} \frac{x^3 - 3x^2 + x}{x^3 - 2x}$$. 2. **Recall the limit rule:** If direct substitution leads to an indeterminate form lik

1. **State the problem:** Find the limit $$\lim_{x \to 2} \frac{3x^2 - 3x}{x - 0}$$. 2. **Rewrite the expression:** The limit is $$\lim_{x \to 2} \frac{3x^2 - 3x}{x}$$.

1. **State the problem:** We need to find the Jacobian of a transformation scaled by the determinant of the matrix \(\begin{bmatrix} 2 & 6 & 5 \end{bmatrix}\) and then evaluate the

1. **Problem statement:** We are given the function $z = \sqrt{3x + 4y}$ and asked to find the rate of change of $z$ at the point $(3,1)$ as $x$ changes while holding $y$ fixed. 2.

1. **State the problem:** Given the function $f(x) = (x^2 + 12)(9 - x^2)$, we need to find critical values, intervals of increase/decrease, local maxima/minima, and intervals of co

1. **Problem statement:** We have the function $$f(x) = x^6 (x - 6)^3$$ defined for $$x \in [-10, 11]$$. We need to find the intervals where $$f$$ is increasing, the region where $

1. **Problem Statement:** Find the points of intersection of the curves $y = x^3$ and $x = y^2$. 2. **Step 1: Express both equations clearly:**

1. **State the problem:** Find the derivative of the function $$f(x) = \frac{\sec x}{\tan x}$$. 2. **Recall the formula:** To differentiate a quotient $$\frac{u}{v}$$, use the quot

1. نبدأ ببيان المسألة: لدينا دالتان $u = x^2 + y^2$ و $v = 2xy$، ونريد إيجاد جهد السرعة $\phi$ الذي يكون ثابتًا مع أحد الخيارات المعطاة. 2. جهد السرعة $\phi$ في الحقل المتجه يُعطى

1. **State the problem:** Find the derivative of the function $f(x) = \frac{1}{\tan^2 x}$. 2. **Rewrite the function:** Note that $\frac{1}{\tan^2 x} = \cot^2 x$. So, $f(x) = \cot^

1. **State the problem:** We need to find real numbers $a$ and $b$ such that the piecewise function $$f(x) = \begin{cases} a x + \frac{b^{x-2}}{x+1} - 1, & x \leq 1 \\ b \sqrt{2x -

1. **Problem:** Evaluate the limit $$\lim_{x \to 2} (x^2 - 4)$$ 2. **Formula and rules:** For limits of polynomial functions, direct substitution is valid because polynomials are c

1. **State the problem:** Find the derivative of the function $y = 8 \sin \sqrt{u}$ with respect to $u$. 2. **Recall the formula:** To differentiate $y = 8 \sin \sqrt{u}$, we use t

1. **Problem statement:** I) Find real values of $a$ and $b$ such that the function

1. **State the problem:** We need to find the derivative of the function $$h(x) = \arctan(\sin(\frac{1}{x^2}))$$. 2. **Recall the chain rule and derivative formulas:**

1. The problem is to find the derivative of a function, but since the function is not specified, let's explain the general process of differentiation. 2. The derivative of a functi