∫ calculus

1. **State the problem:** We need to find the derivative of the function $$h(x) = 2\sqrt{x} + x^3$$ and then evaluate it at $$x=1$$. 2. **Recall the derivative rules:**

1. **State the problem:** We need to find the derivative $h'(2)$ where $h(x) = \frac{f(x)}{2x}$ and $f(x)$ is given by a piecewise linear graph. 2. **Understand the function $f(x)$

1. **State the problem:** We need to find the derivative of $$y = \sin^2(3x)$$ with respect to $$x$$ and then evaluate it at $$x = \frac{\pi}{12}$$. 2. **Formula and rules:** Use t

1. The problem asks to calculate the value of the integral $$\int^{\pi} f(x) \, dx$$. 2. To solve this, we need the explicit function $f(x)$ and the limits of integration. The uppe

1. **State the problem:** We want to prove that $$\int_0^a f(x) dx = \int_0^a f(a-x) dx.$$ 2. **Recall the substitution rule for integrals:** If we let $u = g(x)$ be a differentiab

1. **State the problem:** We need to find the absolute minimum and maximum values of the function $$f(x) = 2.5x^4 + 3x^3 - 2.6x^2 - 5.1x - 5.6$$ on the closed interval $$[-0.5, 1.2

1. **Problem:** Evaluate the integral $$\int \left(2x^3 - 3x^2 + 6x - \frac{3}{x^4} + \frac{2}{x}\right) dx$$. 2. **Formula and rules:** The integral of a sum is the sum of the int

1. **State the problem:** We need to find the value of $t$ at which the revenue of the established department store, represented by $g(t)$, is decreasing at the slowest rate. 2. **

1. **Problem Statement:** We analyze the meaning of the limit and continuity statements given for a function $f$ at points $x_0$.

1. **State the problem:** We want to find the absolute minimum value of the function $$f(x, y) = 5 - 2x^2 - y^2$$ within the rectangular domain defined by $$-2 \leq x \leq 2$$ and

1. **State the problem:** We need to find the critical points of a function. Critical points occur where the derivative of the function is zero or undefined. 2. **Formula and rules

1. The question asks how to determine the second derivative of a function, denoted as $f''(x)$, at each point. 2. The second derivative $f''(x)$ represents the rate of change of th

1. The problem is to find the integral of the function $x \cos(3x) \sin(2x) \csc^3(x)$.\n\n2. We start by writing the integral: $$\int x \cos(3x) \sin(2x) \csc^3(x) \, dx.$$\n\n3.

1. **State the problem:** We need to evaluate the definite integral $$\int_0^1 \sqrt{17 + 4z^2} \, dz.$$\n\n2. **Formula and approach:** This integral involves a square root of a q

1. **State the problem:** Find the derivative of the function $$y = 3 \tan^{-1}\left(x - \sqrt{1 + x^2}\right)$$. 2. **Recall the derivative formula:** The derivative of $$\tan^{-1

1. **State the problem:** We need to find the derivative of the function $$y = \arctan\left(\sqrt{\frac{1 + x}{1 - x}}\right)$$. 2. **Recall the derivative formula:** The derivativ

1. **State the problem:** We want to find constants $a$ and $b$ such that the piecewise function $$f(x) = \begin{cases} ax^3 & x \leq 8 \\ x^2 + b & x > 8 \end{cases}$$

1. **State the problem:** Find the derivative of the function $$f(x) = \frac{x^3 - 9x^2 + 7}{x^2}$$. 2. **Rewrite the function:** Simplify the function by dividing each term in the

1. **State the problem:** Find the limit $$\lim_{x \to 0} \frac{\frac{1}{5+x} - \frac{1}{5}}{x}$$. 2. **Recall the formula:** This is a difference quotient resembling the definitio

1. **State the problem:** We want to evaluate the limit $$\lim_{h \to 0} \frac{\frac{1}{(h + 3)^2} - \frac{1}{9}}{h}$$

1. **State the problem:** We are given the curve $$y = x^{1/2} - 4 + 3x^{-1/2}$$ and need to find: