∫ calculus

1. The problem asks to find the limit of the function $f(x)$ as $x$ approaches $-3$. 2. The limit $\lim_{x \to -3} f(x)$ means we want to find the value that $f(x)$ approaches as $

1. **State the problem:** Find the derivative $h'(x)$ of the function $$h(x) = \frac{1}{60}x^2 - \frac{1}{4}x + \frac{3}{5}$$

1. **State the problem:** We are given a differentiable function $h$ with $h(-5)=10$ and its derivative $h'(x)=2-\sqrt{e^x+2x^2}$. We need to find $h(1)$. 2. **Formula used:** Sinc

1. The problem asks for the average value of the function $$g(x)=4\cos\left(\sqrt{x^2+x+5}\right)$$ on the interval $$[0,5]$$. 2. The formula for the average value of a continuous

1. **State the problem:** We need to sketch a function $f$ with the following limit and value conditions: - $\lim_{x \to 2^+} f(x) = 3$

1. The problem asks for the average value of the function $$f(x) = e^{x^2 - 2x}$$ on the interval $$[-1, 3]$$. 2. The formula for the average value of a function $$f(x)$$ on $$[a,b

1. **State the problem:** We need to find the sum of the areas enclosed by the graphs of $$f(x)=2\cos\left(\frac{\pi x}{4}\right)$$

1. **State the problem:** We want to find how many people left the auditorium between minutes $1$ and $5$ given the rate of change of people remaining is $r(t) = -0.1^t$ people per

1. **State the problem:** Find the area of the region between the graphs of $$f(x)=\frac{4}{x}$$ and $$g(x)=5$$ from $$x=-6$$ to $$x=-2$$. 2. **Formula and rules:** The area betwee

1. The problem is to find the derivative $\frac{dy}{dx}$ of a function $y$ with respect to $x$. 2. To solve this, we need the explicit function $y=f(x)$ or an implicit relation inv

1. مسئله: معادله خط مماس بر نمودار تابع $$f(x) = \sqrt{\frac{x+1}{\sqrt{x}-2}}$$ را در نقطه‌ای که طول مماس برابر 4 است، بیابید. 2. ابتدا باید مشتق تابع $$f(x)$$ را پیدا کنیم تا شیب

1. مسئله: نقطه A روی منحنی $y=3 - x^2$ در ناحیه اول دستگاه مختصات قرار دارد. باید مختصات A را طوری پیدا کنیم که مساحت مثلث OAB بیشینه شود. 2. فرمول مساحت مثلث OAB: مثلث OAB با رئوس

1. مسئله: بررسی مشتق‌پذیری تابع $$f(x) = \begin{cases} x^2 + 3 & x \geq 1 \\ 4x & x < 1 \end{cases}$$

1. The problem asks for the derivative of the function $$y = \ln(x + 3) + \ln 5$$. 2. Recall the derivative rule for natural logarithm: $$\frac{d}{dx}[\ln u] = \frac{1}{u} \cdot \f

1. **Problem Statement:** Given the graph of the derivative function $f'(x)$, classify two different key features on $f'(x)$ and determine the corresponding key features on the ori

1. **Problem statement:** Find the candidate points for extrema and verify the necessary and sufficient conditions for extrema of the function given its derivatives. 2. **Given:**

1. **State the problem:** Find the derivative $f'(x)$ of the function $$y = \ln \left( \frac{(x+1)^4 (x^3+2)}{x-1} \right)$$ using logarithmic differentiation. 2. **Recall the loga

1. **Problem a:** Find the equation of the tangent line to the curve defined by $$x^2 y + y^2 x = 6$$ at the point $$P(1,2)$$. 2. **Formula and rules:** To find the tangent line, w

1. **State the problem:** We need to find the derivative of the function $$f(x) = \sqrt{x} - 2$$. 2. **Recall the formula:** The derivative of $$x^n$$ with respect to $$x$$ is $$nx

1. The problem is to find the inflection points of a function, which are points where the concavity changes. 2. To find inflection points, we use the second derivative test: find w

1. **State the problem:** We need to find the value of the limit $$\lim_{x \to 4} \frac{f(x)}{x^2 - 16}$$ where the function $f(x)$ is given by a graph consisting of semicircles an