∫ calculus

1. **State the problem:** We are given the derivative of a function $f'(x) = -x^3 - 6x^2 - 9x$ and need to find the intervals where the original function $f$ is decreasing. 2. **Re

1. مسئله: بررسی صحت معادله $$\sec x \frac{\partial w}{\partial x} + \sec y \frac{\partial w}{\partial y} = 1$$ برای تابع $$w = \sin y + f(\sin x - \cos y)$$ که در آن $$f$$ تابعی حق

1. The problem is to find the derivative of the function $f(x) = 4 - |x|$ at $x=0$. 2. The absolute value function $|x|$ is defined as:

1. The problem states: "The third derivative is under 16." We need to understand what this means and how to work with it. 2. The third derivative of a function $f(x)$ is denoted as

1. **Problem statement:** Find the Taylor polynomial of degree 3, $T_3(x)$, for $f(x) = \ln(1 + 2x)$ centered at $c=1$, and find an expression for the remainder $R_3(x)$. 2. **Form

1. **State the problem:** Find the area under the curve $y=\frac{11}{x}$ above the x-axis over the interval $[1,2]$. 2. **Set up the integral:** The area $A$ is given by the defini

1. **State the problem:** Find the derivative with respect to $x$ of the function $$f(x) = \frac{x}{2} + x.$$ 2. **Rewrite the function:** $$f(x) = \frac{x}{2} + x = \frac{x}{2} +

1. **State the problem:** Find the derivative of the function $f(x) = \frac{x}{2} + x$ with respect to $x$. 2. **Recall the derivative rules:**

1. **Problem Statement:** Find the left-hand limit, right-hand limit, and the limit at $a=2$ for the function $$f(x) = \frac{x^2 + 3x + 2}{x^2 - 4}$$ 2. **Recall the limit definiti

1. **Problem statement:** Find the derivative $f'(x)$ for $x < 1$ where the function is defined as $$f(x) = x^2 \text{ for } x \leq 1.$$ 2. **Formula used:** The derivative of a fu

1. **State the problem:** Find the right-hand derivative of the piecewise function $$f(x) = \begin{cases} x^2 - 1, & x < 0 \\ 2x - 1, & x \geq 0 \end{cases}$$ at $$x=0$$. 2. **Reca

1. **State the problem:** We need to find the integral of the function $x^2 e^{2x}$ with respect to $x$. 2. **Formula and method:** We will use integration by parts, which states:

1. **Problem Statement:** Differentiate the function $\sin^2 u$, which means find $\frac{d}{du}(\sin^2 u)$. 2. **Formula Used:** Use the chain rule for differentiation: if $y = [f(

1. The problem is to evaluate the expression $e^{\infty^2}$. 2. Recall that $\infty$ represents an unbounded quantity that grows without limit.

1. **State the problem:** Determine the value of $k$ such that the piecewise function $$f(x) = \begin{cases} kx + 3, & x < 1 \\ 8 - x^2, & x \geq 1 \end{cases}$$

1. **Problem Q.4 (i):** Use the Sandwich Theorem to find $$\lim_{x \to 0} f(x)$$ given $$\sqrt{5 - 2x^2} \leq f(x) \leq \sqrt{5 - x^2}$$. 2. The Sandwich Theorem states if $$g(x) \

1. **State the problem:** We are given a function $g(x) = x f(x)$, with $f(3) = 4$ and $f'(3) = -2$. We need to find the equation of the tangent line to the graph of $g$ at $x = 3$

1. **State the problem:** Find the equation of the tangent line to the curve $y = x\sqrt{x}$ that is parallel to the line $y = 1 + 3x$. 2. **Identify the slope of the given line:**

1. **State the problem:** Find the derivative of the function $f(x) = \frac{1}{x^2+1}$ using first principles (the definition of the derivative). 2. **Recall the definition of the

1. **State the problem:** Find the limits:

1. **State the problem:** Find the derivative of the function $f(x) = \frac{1}{x^2} + 1$ using first principles (the definition of the derivative). 2. **Recall the definition of th