1. **Problem 1: Find the instantaneous velocity and acceleration of the drone at $t=5$ given $s(t) = 7t^3 - 3t^2 - 5t + 2$.** 2. The instantaneous velocity is the first derivative of position with respect to time: $$v(t) = s'(t) = \frac{d}{dt}(7t^3 - 3t^2 - 5t + 2) = 21t^2 - 6t - 5$$ 3. Evaluate velocity at $t=5$: $$v(5) = 21(5)^2 - 6(5) - 5 = 21 \times 25 - 30 - 5 = 525 - 35 = 490$$ 4. The acceleration is the derivative of velocity with respect to time: $$a(t) = v'(t) = \frac{d}{dt}(21t^2 - 6t - 5) = 42t - 6$$ 5. Evaluate acceleration at $t=5$: $$a(5) = 42(5) - 6 = 210 - 6 = 204$$ --- 6. **Problem 2: Evaluate $\lim_{(x,y) \to (0,0)} f(x,y)$ for $f(x,y) = \frac{x^2 y}{2 + y^2}$.** 7. Substitute $(0,0)$ directly: $$f(0,0) = \frac{0^2 \times 0}{2 + 0^2} = 0$$ 8. Check limit along $y=0$: $$f(x,0) = \frac{x^2 \times 0}{2 + 0} = 0$$ 9. Check limit along $x=0$: $$f(0,y) = \frac{0 \times y}{2 + y^2} = 0$$ 10. Since the function approaches 0 along different paths, the limit is: $$\lim_{(x,y) \to (0,0)} f(x,y) = 0$$ --- 11. **Problem 3: Compute $\frac{\partial T}{\partial x}$ and $\frac{\partial T}{\partial y}$ at $(1,2)$ for $T(x,y) = 4x^2 y - y^3 + 2$.** 12. Partial derivative with respect to $x$: $$\frac{\partial T}{\partial x} = \frac{\partial}{\partial x}(4x^2 y - y^3 + 2) = 8xy$$ 13. Evaluate at $(1,2)$: $$\frac{\partial T}{\partial x}(1,2) = 8 \times 1 \times 2 = 16$$ 14. Partial derivative with respect to $y$: $$\frac{\partial T}{\partial y} = \frac{\partial}{\partial y}(4x^2 y - y^3 + 2) = 4x^2 - 3y^2$$ 15. Evaluate at $(1,2)$: $$\frac{\partial T}{\partial y}(1,2) = 4 \times 1^2 - 3 \times 2^2 = 4 - 12 = -8$$