∫ calculus

1. **State the problem:** We need to find the derivative $\frac{dy}{dx}$ of the function $$y = (2x^4 - 3)(2x^2 + 1).$$ 2. **Recall the product rule:** For two functions $u(x)$ and

1. **State the problem:** Evaluate the limit $$\lim_{x \to \infty} \frac{x^3}{4x^2 + 3}$$. 2. **Recall the rule for limits at infinity of rational functions:** When evaluating limi

1. **Problem Statement:** Identify points of discontinuity in the given graphs (a), (b), (c) and analyze continuity/discontinuity of given piecewise and domain-restricted functions

1. **Problem statement:** We have two polynomial functions $f(x) = cx^2 + g(x)$ and $g(x)$ with constants $c$ and $k$. Given $g(1) = k$ and $g''(1) = 6$, and the point $(1,5)$ is a

1. We are given the function $f(x) = x^3 - 6x^2$ and asked to find the interval where it is convex downward (concave up). 2. To determine concavity, we use the second derivative te

1. **Stating the problem:** Determine on which interval the function $f(x) = x^2$ is convex downward (concave up). 2. **Recall the definition:** A function is concave up on an inte

1. Let's start by stating the problem: You want to know how to decide which integration technique to use, such as partial fractions or others, and how the power of terms influences

1. **Problem Statement:** We are given that the first derivative $k'(c) = 0$ and the second derivative $k''(c) = 0$ at the point $x = c$. We want to determine what can be concluded

1. **Problem Statement:** Determine whether the points (0, 2) and (1, e + 1/e) are local minima or maxima of the function $$f(x) = \frac{e^{2x} + 1}{e^x}$$. 2. **Rewrite the functi

1. **State the problem:** We are given the function $f(x) = \frac{e^{2x+1}}{e^x}$ and asked to determine whether the points $(0,2)$ and $(1, e + \frac{1}{e})$ are local minima or m

1. **State the problem:** We have a function $$f(x) = 3ax^3 - bx - 5$$ and it is given that there is a local maximum at $$x=1$$. 2. **Recall the condition for local maxima:** At a

1. **State the problem:** We need to find the largest open intervals where the function $$f(x) = 3x^3 + 8x^2 - 7x + 3$$ is concave upward or concave downward, and find any points o

1. **State the problem:** We are given the function $$f(x) = \frac{x^2}{2+x}$$ and need to find its second derivative $$f''(x)$$. Then, evaluate $$f''(0)$$ and $$f''(5)$$ or determ

1. **State the problem:** We are given the function $f(x) = 3x e^{-4x}$ and need to find the location and value of any local maxima and minima. 2. **Recall the formula and rules:**

1. **State the problem:** We need to use the first derivative test to find the local extrema (maximum and minimum) of the function $$f(x) = 3x^2 - 4x$$ and determine their location

1. **State the problem:** We need to find the intervals where the function $$f(x) = 2x^3 + 6x^2 - 18x + 3$$ is increasing or decreasing. 2. **Formula and rules:** To determine wher

1. **State the problem:** Solve the differential equation $$\frac{dy}{dx} = \frac{\sin(2x)}{\cos(y) \sin^7(y)}.$$\n\n2. **Rewrite the equation:** This is a separable differential e

1. **بيان المسألة:** لدينا دالة $f$ معرفة على $\mathbb{R}$، متصلة ومشتقة تابعياً، وتحقق المعادلة التفاضلية $f'(x) = f(x) + x - 1$. المطلوب: (1) إثبات أن $f(x) = (x-1)e^x + 1$.

1. **Problem Statement:** We need to evaluate the latency for four Azure Availability Zones using given limit formulas as $x \to 6$ and rank them by latency. Then, evaluate two add

1. **Problem Statement:** We have two functions: $$f(n) = n \ln\left(\frac{10}{n}\right)$$

1. **Problem Statement:** We are given cost functions $f(x)$ representing the cost of producing $x$ cell phones during summer and winter periods. We need to interpret the meaning o