∫ calculus

1. The problem asks to identify the graph of a continuous function $f$ such that: - $f(0) = 3$

1. **Problem:** Evaluate $$\int \frac{dx}{\sqrt{(x-\alpha)(\beta - x)}}$$ where $$\beta > \alpha$$. Step 1: Use substitution $$x = \alpha + (\beta - \alpha) \sin^2 \theta$$.

1. **State the problem:** We are given the function for the number of words remembered after $t$ days: $$w(t) = 100 \times (1 - 0.1t)^2, \quad 0 \leq t \leq 10.$$ We need to find t

1. The problem asks us to find and simplify the difference quotient $$\frac{f(a+h)-f(a)}{h}$$ for the function $$f(x) = \cos x$$. 2. Substitute the function into the difference quo

1. **State the problem:** We want to find the number of points where the function $$f(x) = \begin{cases} \min(1, x^2, x^3), & x < 1 \\ \min(x^3, 3x - 2), & x \geq 1 \end{cases}$$

1. Problem: Given the equation $x^2 + y^2 = 16$, find $\frac{\partial y}{\partial x}$. Step 1: Differentiate both sides with respect to $x$ implicitly.

1. **State the problem:** We have a piecewise function $$f(x) = \begin{cases} \frac{x^3 - a x^2 + 2}{x^2 - 3x + 2} & 0 < x < 1 \\ b^2 x^2 + b x + 1 & x = 1 \\ \left(1 + (\ln c) \ta

1. Problem: Given $f(x) = \frac{2x+3}{x-1}$, find its inverse, domain, range, and verify compositions. 1. Find $f^{-1}(x)$:

1. Problem: Find $\frac{dz}{dt}$ if $z = x^2 y + \sin y$, with $x = t^2$ and $y = \ln t$. Step 1: Express $z$ in terms of $t$ using given substitutions.

1. Find $f_x$ and $f_y$ if $f(x,y) = x^3 y^2 + 4x$. Step 1: Identify the function: $f(x,y) = x^3 y^2 + 4x$.

1. **State the problem:** Evaluate the triple integral $$\int_{\frac{\pi}{2}}^{5} \int_0^0 \int_0^0 r \sin \theta \sec^2 \phi \cos \theta \, dr \, d\theta \, d\phi.$$ 2. **Analyze

1. Find $f_x$ and $f_y$ if $f(x,y) = x^3 y^2 + 4x$. Step 1: Identify the function: $f(x,y) = x^3 y^2 + 4x$.

1. **Problem:** Given $x = \frac{1 - t^2}{1 + t^2}$ and $y = \frac{2t}{1 + t^2}$, prove that $$\frac{dy}{dx} + \frac{x}{y} = 0.$$\n\n2. **Find derivatives $\frac{dx}{dt}$ and $\fra

1. **Find the derivatives:** 1.1. Find $D_x (\ln 3 \sqrt[3]{x})$.

1. **State the problem:** Find the equation of the tangent line to the curve $$y = x^3 - 6x^2 + 9x + 4$$ at the point where $$x = 2$$. 2. **Find the derivative:** The derivative $$

1. Problem (iii): Evaluate $$\int (\sqrt{x} + \frac{1}{\sqrt{x}})^2 \, dx$$ Step 1: Expand the integrand:

1. The problem is to evaluate the definite integral $$\int_0^1 x \ln(x+1) \, dx.$$\n\n2. Use integration by parts. Let \(u = \ln(x+1)\) and \(dv = x \, dx\). Then \(du = \frac{1}{x

1. **بيان المسألة:** لدينا دالتان:

1. The problem asks us to explain why Rolle's theorem does not apply to the function $f(x) = \frac{1}{x} - 3$ on the interval $[-4,4]$. 2. Rolle's theorem states that if a function

1. **Problem a:** Find $$\lim_{x \to 0} \frac{\sin x}{x}$$ using Bernoulli-l’Hôpital’s rule. Since direct substitution gives $$\frac{0}{0}$$, apply l’Hôpital’s rule:

1. The problem is to find the derivative of the function $f(x) = e^{2 - x^2}$.\n\n2. Recall the chain rule for derivatives: if $f(x) = e^{g(x)}$, then $f'(x) = e^{g(x)} \cdot g'(x)