∫ calculus

1. **State the problem:** Find the acute angle between the curves $y_1 = x^2$ and $y_2 = 2x + 3$ at their point of intersection where $x = 1$. 2. **Find the point of intersection:*

1. **Problem:** Use the Intermediate Value Theorem (IVT) to show that the function $$f(x) = e^{x/4} + \frac{1}{8}x^2 - 4$$

1. **State the problem:** Evaluate the integral $$\int \frac{\sin 3x}{1 + \cos 3x} \, dx.$$\n\n2. **Recall the formula and rules:** We can use the substitution method and trigonome

1. Problem: Find the first partial derivatives of $$z = x^3 - 3x^2 y^4 + y^2$$ with respect to $$x$$ and $$y$$. 2. Formula: The partial derivative of $$z$$ with respect to $$x$$ is

1. **Problem Statement:** Approximate the integral $$\int_0^{0.35} \frac{2}{x^2 - 4} \, dx$$ using three numerical methods: (a) Gaussian quadrature with $n=3$, (b) Closed Newton-Co

1. **Problem statement:** Evaluate the integral $$\int 2x(x^2 + 4)^3 \, dx$$ using the substitution $$u = x^2 + 4$$. 2. **Formula and substitution:** We use substitution for integr

1. **Problem:** Find the volume of the solid of revolution formed by revolving the region bounded by $y = x^2$ and $x = y^2$ about the line $y = 1$. 2. **Identify the curves and re

1. The problem asks to select the correct options for given integration and calculus questions. 2. (i) If $f$ is integrable, it must be continuous almost everywhere, so the best ch

1. The problem asks to identify the function $f$ and the number $a$ for which the limit $$\lim_{h \to 0} \frac{\sqrt[5]{32 + h} - \sqrt[5]{32}}{h}$$

1. **State the problem:** Calculate the limit $$\lim_{x \to 1} \frac{\sqrt{3x - 1} - \sqrt{3 - x}}{x - 1}$$

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 $f(x)$