1. **Problem:** Express the volume $V$ of a cube as a function of the area $A$ of its base. Step 1: The base of a cube is a square with side length $s$. Step 2: Area of base $A = s^2$. Step 3: Volume of cube $V = s^3$. Step 4: Express $s$ in terms of $A$: $s = \sqrt{A}$. Step 5: Substitute into volume: $$V = (\sqrt{A})^3 = A^{3/2}$$. 2. **Problem:** Evaluate $$\lim_{x \to 2} \frac{\sqrt{x} - \sqrt{2}}{x - 2}$$ using algebraic techniques. Step 1: Direct substitution gives $\frac{0}{0}$, an indeterminate form. Step 2: Multiply numerator and denominator by conjugate $\sqrt{x} + \sqrt{2}$: $$\frac{\sqrt{x} - \sqrt{2}}{x - 2} \times \frac{\sqrt{x} + \sqrt{2}}{\sqrt{x} + \sqrt{2}} = \frac{x - 2}{(x - 2)(\sqrt{x} + \sqrt{2})} = \frac{1}{\sqrt{x} + \sqrt{2}}$$ Step 3: Now substitute $x=2$: $$\frac{1}{\sqrt{2} + \sqrt{2}} = \frac{1}{2\sqrt{2}} = \frac{\sqrt{2}}{4}$$. 3. **Problem:** Express the limit $$\lim_{n \to +\infty} \left(1 + \frac{1}{n} \right)^n$$ in terms of $e$. Step 1: This is the classic definition of Euler's number $e$. Step 2: Therefore, $$\lim_{n \to +\infty} \left(1 + \frac{1}{n} \right)^n = e$$. 4. **Problem:** Find $\frac{dy}{dx}$ if $$x^2 - 4xy - 5y = 0$$. Step 1: Differentiate both sides implicitly w.r.t. $x$: $$2x - 4(y + x \frac{dy}{dx}) - 5 \frac{dy}{dx} = 0$$. Step 2: Expand: $$2x - 4y - 4x \frac{dy}{dx} - 5 \frac{dy}{dx} = 0$$. Step 3: Group $\frac{dy}{dx}$ terms: $$-4x \frac{dy}{dx} - 5 \frac{dy}{dx} = 4y - 2x$$. Step 4: Factor: $$\frac{dy}{dx}(-4x - 5) = 4y - 2x$$. Step 5: Solve for $\frac{dy}{dx}$: $$\frac{dy}{dx} = \frac{4y - 2x}{-4x - 5} = \frac{2x - 4y}{4x + 5}$$. 5. **Problem:** Differentiate w.r.t. $\theta$: $$\tan^3 \theta \sec^2 \theta$$. Step 1: Let $f(\theta) = \tan^3 \theta \sec^2 \theta$. Step 2: Use product rule: $$f' = 3 \tan^2 \theta \sec^2 \theta \cdot \sec^2 \theta + \tan^3 \theta \cdot 2 \sec^2 \theta \tan \theta$$ Step 3: Simplify: $$= 3 \tan^2 \theta \sec^4 \theta + 2 \tan^4 \theta \sec^2 \theta$$. 6. **Problem:** Find $\frac{dy}{dx}$ if $$y = \ln(9 - x^2)$$. Step 1: Use chain rule: $$\frac{dy}{dx} = \frac{1}{9 - x^2} \cdot (-2x) = \frac{-2x}{9 - x^2}$$. 7. **Problem:** Evaluate indefinite integral: $$\int (3x^2 - 2x + 1) \, dx$$. Step 1: Integrate term-wise: $$\int 3x^2 dx = x^3$$ $$\int -2x dx = -x^2$$ $$\int 1 dx = x$$ Step 2: Combine and add constant $C$: $$x^3 - x^2 + x + C$$. 8. **Problem:** Evaluate integral by parts: $$\int x^2 \ln x \, dx$$ Step 1: Let $u = \ln x$ (differentiable for $x>0$), $dv = x^2 dx$. Step 2: Then $du = \frac{1}{x} dx$, $v = \frac{x^3}{3}$. Step 3: Apply integration by parts formula: $$\int u \, dv = uv - \int v \, du$$ Step 4: Substitute: $$= \frac{x^3}{3} \ln x - \int \frac{x^3}{3} \cdot \frac{1}{x} dx = \frac{x^3}{3} \ln x - \frac{1}{3} \int x^2 dx$$ Step 5: Integrate: $$= \frac{x^3}{3} \ln x - \frac{1}{3} \cdot \frac{x^3}{3} + C = \frac{x^3}{3} \ln x - \frac{x^3}{9} + C$$. 9. **Problem:** Evaluate definite integral: $$\int_{\pi/6}^{\pi/3} \cos t \, dt$$. Step 1: Integrate: $$\int \cos t dt = \sin t + C$$ Step 2: Evaluate limits: $$\sin \left(\frac{\pi}{3}\right) - \sin \left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} - \frac{1}{2}$$. 10. **Problem:** Solve differential equation: $$\frac{dy}{dx} = \frac{1 - x}{y}$$. Step 1: Separate variables: $$y dy = (1 - x) dx$$ Step 2: Integrate both sides: $$\int y dy = \int (1 - x) dx$$ Step 3: Compute integrals: $$\frac{y^2}{2} = x - \frac{x^2}{2} + C$$ Step 4: Multiply both sides by 2: $$y^2 = 2x - x^2 + C'$$ where $C' = 2C$. 11. **Problem:** Graph solution set of linear inequality: $$5x - 4y \leq 20$$. Step 1: Rewrite inequality: $$-4y \leq 20 - 5x$$ Step 2: Divide by $-4$ (reverse inequality): $$y \geq \frac{5x}{4} - 5$$ Step 3: The graph is the half-plane above or on the line $y = \frac{5}{4}x - 5$. 12. **Problem:** Define objective function and optimal solution. Step 1: An objective function is a function to be maximized or minimized in optimization problems. Step 2: An optimal solution is the input value(s) that maximize or minimize the objective function subject to constraints. 13. **Problem:** Given vectors $$u = i + 2j - k, \quad v = 3i - 2j + 2k, \quad w = 5i - j + 3k$$ Find $$|3v + w|$$. Step 1: Compute $3v + w$: $$3v = 9i - 6j + 6k$$ $$3v + w = (9 + 5)i + (-6 - 1)j + (6 + 3)k = 14i - 7j + 9k$$ Step 2: Compute magnitude: $$|3v + w| = \sqrt{14^2 + (-7)^2 + 9^2} = \sqrt{196 + 49 + 81} = \sqrt{326}$$. 14. **Problem:** Prove $$a \times (b + c) + b \times (c + a) + c \times (a + b) = 0$$. Step 1: Use distributive property of cross product: $$a \times b + a \times c + b \times c + b \times a + c \times a + c \times b$$ Step 2: Group terms: $$(a \times b + b \times a) + (a \times c + c \times a) + (b \times c + c \times b)$$ Step 3: Since $x \times y = - y \times x$, each pair sums to zero: $$0 + 0 + 0 = 0$$. 15. **Problem:** Prove $$u \cdot (v \times w) + v \cdot (w \times u) + w \cdot (u \times v) = 3 u \cdot (v \times w)$$. Step 1: Recall scalar triple product cyclic property: $$u \cdot (v \times w) = v \cdot (w \times u) = w \cdot (u \times v)$$ Step 2: Sum of three equal terms: $$3 u \cdot (v \times w)$$. **Final answers:** 1. $V = A^{3/2}$ 2. $\frac{\sqrt{2}}{4}$ 3. $e$ 4. $\frac{dy}{dx} = \frac{2x - 4y}{4x + 5}$ 5. $3 \tan^2 \theta \sec^4 \theta + 2 \tan^4 \theta \sec^2 \theta$ 6. $\frac{dy}{dx} = \frac{-2x}{9 - x^2}$ 7. $x^3 - x^2 + x + C$ 8. $\frac{x^3}{3} \ln x - \frac{x^3}{9} + C$ 9. $\frac{\sqrt{3}}{2} - \frac{1}{2}$ 10. $y^2 = 2x - x^2 + C$ 11. Region $y \geq \frac{5}{4}x - 5$ 12. Objective function: function to optimize; Optimal solution: input maximizing/minimizing it. 13. $|3v + w| = \sqrt{326}$ 14. Identity holds: sum equals zero. 15. Identity holds: sum equals $3 u \cdot (v \times w)$.