∫ calculus

1. The problem is to evaluate the limit: $$\lim_{x \to 0} \frac{du(x+h) - du(h)}{x}$$

1. **Problem:** Evaluate $$\int \frac{4x + 2}{x^3 + x^2 - 2x} \, dx$$ using partial fractions. Step 1: Factor the denominator:

1. Problem b) Find the value of $p$ for which the function $$f(x) = \begin{cases} \frac{\sqrt{1+px} - \sqrt{1-px}}{x}, & -1 \leq x < 0 \\ \frac{2x+1}{x-2}, & 0 \leq x \leq 1 \end{c

1. **Find the derivative of** $f(x) = \sqrt[3]{x^2} + \frac{3x}{x^4} - \frac{1}{2 \sqrt[4]{x^5}} - 6$. Do not simplify. Step 1: Rewrite the function using exponents:

1. المشكلة المعطاة هي مقارنة الميل بين منحنيين ن(س) ون'(س). 2. معطى أن ميل منحنى الانتقال ن(س) لأي نقطة يساوي 3.

1. The problem asks to find the limit as $n$ approaches infinity of the expression $\frac{n + 8}{n + 4}$.\n\n2. To analyze the limit, divide both numerator and denominator by $n$:

1. **Understand the problem:** We are asked to order the derivative values (slopes) of the function $f$ at points A, B, C, and D from smallest to largest.

1. في السؤال المطروح، نريد حساب النهاية $$\lim_{x \to 1} \frac{f(x)}{x-1}$$. 2. هذه النهاية تمثل ميل المماس للدالة عند النقطة $$x=1$$ إذا كانت النهاية موجودة، وهذا يتشابه مع تعريف

1. The problem asks us to determine the interval on which the function $f(x) = e^x$ is increasing. 2. Recall that $f(x) = e^x$ is an exponential function where the base $e$ is Eule

1. Statement of the problem: For the curve $y = x^3 + 5x^2 + P x - 18$ the gradient at $x = -4$ is $-15$. Find (a)(i) the value of $P$, (ii) the equation of the normal at $x = -1$,

1. **State the problem:** We know for a function $p(x)$, that the first derivative at $x=5$ is zero, i.e., $p'(5) = 0$, and the second derivative at $x=5$ is negative, i.e., $p''(5

1. المشكلة: لدينا د(س) = |س - 1|، ونريد إيجاد الاقتران المتكامل (التي هي دالة تكامل لد(س)) على الفترة [0,2]. 2. بدايةً، نلاحظ أن د(س) = |س - 1| يعني أن د(س) = 1 - س عندما س \leq 1،

1. We are asked to find the inflection points of the function $g(x) = x^4 - 6x^2$. 2. Inflection points occur where the second derivative changes sign, so first find the first deri

1. **State the problem:** Find the x-coordinate(s) of the inflection point(s) of the function $$g(x) = \frac{1}{x^2 + 1}$$. 2. **Recall inflection points:** Inflection points occur

1. The problem asks us to find the intervals where the function $f(x) = x^4 - 24x^2 + 4$ is convex downward. Convex downward means the graph is concave down, which occurs where the

1. The problem asks us to find where the function $k(x) = \frac{1}{x^2 + 3}$ is convex downward. 2. Recall that a function is convex downward where its second derivative is negativ

1. The problem is to find the limit $$\lim_{x \to 7} \frac{x^2 - 4x - 21}{x^2 - 49}.$$\n\n2. Substitute $x = 7$ directly to check if the expression is defined:\n$$\frac{7^2 - 4(7)

1. The problem asks to determine whether certain points given are local minima or maxima of the function 2. The function is $f(x) = \frac{e^{2x+1}}{e^x}$. Simplify the function fir

1. Problem: Evaluate the integral $$\int_{\pi/4}^{\pi/2} \sin x \frac{d}{dx} + \cos x F(X) \, dx$$. 2. Interpretation: The integral as written is ambiguous. It seems to involve $$\

1. لنبدأ بفهم المعطيات: لدينا معلومة $\cos 2 = 3\sqrt{2}$ وقيمة أخرى $\frac{\pi}{4} = 2$. 2. المطلوب هو إيجاد قيمة التكامل $$\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} (\sin x + \cos x \

1. لنفهم المسألة، لدينا دالة ن حيث ن(3) = 3\sqrt{2} ون(\frac{\pi}{2}) = 2\frac{\pi}{4} = \frac{\pi}{2}. 2. المطلوب حساب التكامل المحدود