∫ calculus

1. **State the problem:** Find where the function $f(x) = \sqrt{x^2 + 9}$ has horizontal and vertical tangents. 2. **Recall the formula for the derivative:** To find tangents, we n

1. **Problem Statement:** Find the horizontal and vertical tangents of the function $f(x) = \sqrt{x^2 + 9}$. 2. **Formula and Rules:** The derivative $f'(x)$ gives the slope of the

1. **State the problem:** We are given the implicit equation $$xy^2 - 2xy + x^2y = 12$$ and need to find the slope of the tangent line at the point $$(1,4)$$. 2. **Recall the formu

1. **Problem:** Find the values of $a$ such that the limit $$\lim_{x \to 0} \frac{\sin 2x + a \sin x}{x^3}$$ exists, and then find the limit. 2. **Formula and rules:** Use Taylor e

1. لنبدأ بتوضيح معنى "معدلات" في الرياضيات، وهي عادة تشير إلى المشتقات أو التغيرات في الكميات. 2. إذا كنت تقصد حساب المشتقة لدالة معينة، فالمشتقة تعبر عن معدل التغير اللحظي للدالة.

1. **State the problem:** We need to find the derivative $\frac{dy}{dx}$ of the function $y = 10^{1 - x^2}$. 2. **Recall the formula:** For a function of the form $y = a^{u(x)}$, t

1. The problem is to analyze the function for intervals of increase, decrease, asymptotes, and find the y-intercept. 2. To determine where a function is increasing or decreasing, w

1. **State the problem:** We need to find the intervals where the function $$f(x) = x^4 - 6x^3 - 14$$ is increasing. 2. **Recall the rule:** A function is increasing where its deri

1. The problem asks to determine the intervals where the function $f(x)$ is decreasing given the graph of its derivative $f'(x)$. 2. Important rule: A function $f(x)$ is decreasing

1. The problem asks to determine the intervals where the function $f(x)$ is increasing given the graph of its derivative $f'(x)$. 2. Recall that a function $f(x)$ is increasing whe

1. **Problem Statement:** Given the graph of the derivative $f'(x)$, determine the intervals where the original function $f(x)$ is increasing. 2. **Key Concept:** A function $f(x)$

1. The problem is to find the absolute minimum of a function on the closed interval $[0,4]$. 2. To find the absolute minimum on a closed interval, we evaluate the function at criti

1. We are given the derivative of a function: $f'(x) = x^2 - 3 - 3 \sin(2x^2 - x)$ and the domain $x \in [0,4]$. 2. The problem is to understand or analyze this derivative function

1. The problem gives the derivative of a function as $f'(x) = (x^2 - 2) \sin(x)$ and an initial condition $f(-1) = 8$. We are asked to find the original function $f(x)$ on the inte

1. **Problem Statement:** Given the derivative of a function $f'(x) = (x^2 - 2) \sin(x)$ and the initial condition $f(-1) = 8$, find the original function $f(x)$. 2. **Formula and

1. **State the problem:** We are given the derivative of a function $$f'(x) = x^{2} \sin(x)$$ and the value $$f(-1) = -6$$. We need to find the absolute minimum value of $$f$$ on t

1. We are given the derivative of a function: $f'(x) = x^2 \sin(3x)$ and the initial condition $f(0) = 5$. We want to find the original function $f(x)$ on the interval $[-2, 2]$. 2

1. **State the problem:** We need to find the intervals where the function $f(x) = x^4 - 4x^3$ is decreasing. 2. **Recall the rule:** A function is decreasing where its derivative

1. **State the problem:** We are given the function $$f(x) = -x^4 - 12x^3 - 36x^2$$ and need to find the intervals where $$f$$ is increasing. 2. **Recall the rule:** A function is

1. **State the problem:** We need to find the intervals where the function $f(x) = x^3 - 21x$ is increasing. 2. **Recall the rule:** A function is increasing where its derivative $

1. **State the problem:** We need to find the intervals where the function $$f(x) = -x^3 + 12x^2 - 36x$$ is increasing. 2. **Recall the rule:** A function is increasing where its d