∫ calculus

1. **State the problem:** We are given the function $f(x) = \ln(13x^2 + 5)$ and need to find the points of inflection $Q_1$ and $Q_2$ where the concavity changes. 2. **Find the fir

1. **State the problem:** We have a function $$f(x) = \frac{x^2}{3 + 3x}$$ and its second derivative is given by $$f''(x) = \frac{Q_1}{(3 + 3x)^3}$$ where $$Q_1$$ is a constant. We

1. **Problem:** Find the derivative of $$y = 5^{2x} \sin^2 x$$. 2. **Formula and rules:** Use the product rule: $$\frac{d}{dx}[u v] = u' v + u v'$$.

1. **Problem statement:** Given the function $f(x) = \frac{7}{2 + 4x^2}$, find its power series representation, interval of convergence, series for its derivative, series for its d

1. **Problem Statement:** We are given a profit function for a production division: $$P(x) = \frac{500 \ln(x + 1)}{(x + 1)^2}$$

1. **State the problem:** We are given the marginal revenue function $R'(q) = 100 q^{-\frac{1}{2}}$ and the marginal cost function $C'(q) = 0.4q$. We know the total profit at $q=16

1. The problem is to evaluate the integral $$\int_{}^{x} \frac{1}{\cos^2(t+\frac{\pi}{2})} \, dt$$. 2. Recall the trigonometric identity: $$\cos\left(\theta + \frac{\pi}{2}\right)

1. نبدأ بكتابة الدالة المعطاة: $$F(x) = 2x + 1 - x e^{-x}$$ 2. المطلوب هو حساب النهاية عندما يقترب $x$ من $-\infty$.

1. المشكلة: كيفاش نتعامل مع تعبير فيه ناقص مالانهاية. 2. في الرياضيات، ناقص مالانهاية ($-\infty$) تعني قيمة تقترب من سالب عدد كبير جداً بلا حدود.

1. المشكلة غير مكتملة، ولكن سأفترض أنك تسأل عن كيفية حساب النهاية عند نقطة معينة لدالة. 2. لحساب النهاية عند نقطة $a$ لدالة $f(x)$، نستخدم التعريف:

1. **Statement of the problem:** Given the function $$f(x) = 2x + 1 - xe^{-x}$$, calculate the limits $$\lim_{x \to +\infty} f(x)$$ and $$\lim_{x \to -\infty} f(x)$$.

1. **State the problem:** Find the derivative of the function $f(x) = 5 - x^3$ using the limit definition of the derivative. 2. **Recall the limit definition of the derivative:**

1. The problem is to find the derivative of the expression $(xy)' + (x' + y')$. 2. Recall the product rule for derivatives: $\frac{d}{dx}(uv) = u'v + uv'$, where $u$ and $v$ are fu

1. **Problem:** Calculate the limit $$\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^{x+2}$$ **Step 1:** Recall the important limit definition of the number $e$:

1. **State the problem:** Find the limit $$\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^{x+2}$$. 2. **Recall the important formula:** The expression $$\left(1 + \frac{1}{x}\rig

1. **Problem statement:** Calculate the limits of the function $$f(x) = (-x^3 + 2x^2) e^{-x+1}$$

1. The problem states: "Let's say the limit is -3." However, this is incomplete as a limit problem requires a function and a point to evaluate the limit at. 2. To solve a limit pro

1. Let's start by stating the problem: We want to construct a table to explore the limit of a function as it approaches a point where the limit is negative. 2. Consider the functio

1. **Problem statement:** Test the convergence or divergence of the series \(\sum_{n=1}^\infty \frac{(-1)^{n+1}}{(2n+1)!} \). 2. **Formula and rules:** For series with factorials a

1. **Stating the problem:** Evaluate the improper integral $$\int_0^\infty 7e^x \, dx$$. 2. **Formula and rules:** The integral of an exponential function $$e^x$$ is $$e^x$$ itself

1. **Problem Statement:** Evaluate the following limits using exponential and logarithmic concepts without L'Hôpital's rule: (a) $$\lim_{x \to \frac{\pi}{2}} (\tan x)^{\frac{\pi}{2