∫ calculus

1. **Problem Statement:** We are given a curve $\hat{f}(x)$ and asked to identify which of the statements (a) to (d) about the function $f$ are correct, except one. 2. **Understand

1. **Problem statement:** We have a function $f$ defined on the interval $[a,b]$ with values in $\mathbb{R}^-$ (meaning $f(x) < 0$ for all $x \in [a,b]$). We want to determine for

1. **Problem Statement:** We are given the graph of the first derivative $f'(x)$ of a continuous function $f$ on $\mathbb{R}$. We need to identify which statement among (a), (b), (

1. The problem asks us to determine the correct relationship between the function $f(x)$ and the tangent line $g(x)$ to the curve $y=f(x)$ at any point $(x,y)$. 2. Recall that the

1. **Problem Statement:** Given that $f'(x) < 0$ and $f''(x) > 0$ for each $x \in [a,b]$, determine which graph represents the function $f$ on the interval $[a,b]$. 2. **Understand

1. **Problem Statement:** We analyze the function $f$ defined on the interval $[1,5]$ and determine which of the given statements (a) to (d) about its critical points, inflection p

1. **Problem Statement:** Given the graph of the second derivative $f''(x)$, determine which statement about the function $f$ is true. 2. **Understanding the problem:** The graph s

1. **Problem Statement:** We are given the graph of the derivative function $f'(x)$ of a continuous function $f$ on $\mathbb{R}$. The graph is a parabola opening upward with vertex

1. **Problem:** Evaluate the limit $$\lim_{x \to 2} \frac{\sqrt{6 - x^2}}{3 - x - 1}$$ 2. **Formula and rules:** To evaluate limits involving square roots and rational expressions,

1. **مسئله:** محاسبه مشتق تابع داده شده است. 2. برای حل مشتق، ابتدا باید تابع مورد نظر را مشخص کنید. لطفاً تابعی که می‌خواهید مشتق بگیرید را ارسال کنید.

1. **State the problem:** Find the third derivative $y^{(3)}$ of the function $$y = \frac{\lg(x - 1)}{x^3}$$ where $\lg$ denotes the base-10 logarithm. 2. **Recall the formula and

1. **Problem statement:** We have an isosceles triangle with base length $20\sqrt{3}$ cm. The two equal legs decrease at a rate of 3 cm/h. We want to find the rate at which the tri

1. Let's state the problem: Optimization involves finding the maximum or minimum value of a function, often subject to certain constraints. 2. The general approach uses the derivat

1. **State the problem:** Find all points on the curve defined by $$x^2 + y^2 + 4x - 2y = -1$$ where the tangent lines are (a) horizontal and (b) vertical. 2. **Rewrite the curve e

1. The problem is to find the derivative of a function, but the function is not specified. Please provide the function to differentiate.

1. **State the problem:** Find the derivative of the function $g(x) = 3e^x \cos x$. 2. **Formula used:** To differentiate a product of two functions, use the product rule:

1. **State the problem:** Find the equation of the tangent line to the curve $f(x) = 5e^x + 2\sin x$ at $x=0$. 2. **Recall the formula for the tangent line:** The equation of the t

1. **State the problem:** Find the derivative of the function $$y = \frac{5x^4 + 3x - 5 + 2}{x^3 + \pi x}$$. 2. **Recall the formula:** For a function $$y = \frac{u(x)}{v(x)}$$, th

1. **Problem Statement:** Find the equation of the tangent line to the curve at the given $x$ value. 2. **Formula and Rules:** The equation of the tangent line at $x=a$ is given by

1. **State the problem:** We are given the displacement function $s(t) = 4t^3 + 3t^2$ and need to find the instantaneous rate of change of displacement at $t=3$ seconds. This rate

1. **State the problem:** Find the $x$-value on the curve $y = x^2$ where the tangent line is parallel to the line $y = 2x - 4$. 2. **Identify the slope of the given line:** The li