1. **Problem 1:** Find the derivative $\frac{dy}{dx}$ of the implicit function given by $$x^2 y - 5y = 12$$ and evaluate it at the point $(1, -3)$. 2. **Step 1:** Differentiate both sides with respect to $x$. Use the product rule for $x^2 y$ and the chain rule for $y$. $$\frac{d}{dx}(x^2 y) - \frac{d}{dx}(5y) = \frac{d}{dx}(12)$$ 3. Applying the product rule: $$\frac{d}{dx}(x^2 y) = x^2 \frac{dy}{dx} + 2x y$$ 4. Differentiating the other terms: $$\frac{d}{dx}(5y) = 5 \frac{dy}{dx}$$ $$\frac{d}{dx}(12) = 0$$ 5. Substitute back: $$x^2 \frac{dy}{dx} + 2x y - 5 \frac{dy}{dx} = 0$$ 6. Group terms with $\frac{dy}{dx}$: $$x^2 \frac{dy}{dx} - 5 \frac{dy}{dx} = -2x y$$ 7. Factor out $\frac{dy}{dx}$: $$\frac{dy}{dx} (x^2 - 5) = -2x y$$ 8. Solve for $\frac{dy}{dx}$: $$\frac{dy}{dx} = \frac{-2x y}{x^2 - 5}$$ 9. Evaluate at the point $(1, -3)$: $$\frac{dy}{dx} = \frac{-2(1)(-3)}{1^2 - 5} = \frac{6}{-4} = -\frac{3}{2}$$ --- 1. **Problem 2:** A particle moves along a horizontal line with position function $$s(t) = t^3 - 9t^2 + 15t + 4$$ for $t \geq 0$. Find when the particle is speeding up. 2. **Step 1:** Speeding up means velocity and acceleration have the same sign. 3. Find velocity $v(t)$ by differentiating position: $$v(t) = \frac{ds}{dt} = 3t^2 - 18t + 15$$ 4. Find acceleration $a(t)$ by differentiating velocity: $$a(t) = \frac{dv}{dt} = 6t - 18$$ 5. Determine intervals where $v(t)$ and $a(t)$ have the same sign. 6. Solve $v(t) = 0$: $$3t^2 - 18t + 15 = 0 \Rightarrow t^2 - 6t + 5 = 0$$ $$t = \frac{6 \pm \sqrt{36 - 20}}{2} = \frac{6 \pm 4}{2}$$ $$t = 1 \text{ or } 5$$ 7. Solve $a(t) = 0$: $$6t - 18 = 0 \Rightarrow t = 3$$ 8. Test intervals for signs: - For $t \in [0,1)$: $v(0) = 15 > 0$, $a(0) = -18 < 0$ (different signs, not speeding up) - For $t \in (1,3)$: $v(2) = 3(4) - 36 + 15 = -9 < 0$, $a(2) = -6 < 0$ (same sign, speeding up) - For $t \in (3,5)$: $v(4) = 3(16) - 72 + 15 = -9 < 0$, $a(4) = 6 > 0$ (different signs, not speeding up) - For $t > 5$: $v(6) = 3(36) - 108 + 15 = 15 > 0$, $a(6) = 18 > 0$ (same sign, speeding up) 9. **Answer:** The particle is speeding up on intervals $$ (1, 3) \cup (5, \infty) $$.