Saturday, 5 September 2026

Show that \( K_1 \cdot (K_2 \times K_3) = 1/[V_1 \cdot (V_2 \times V_3)] \).

Find $\int (t+1)(t-1) dt$.

\[\begin{aligned}d_{n} &= \frac{192 L^{2}}{(2 n-1)^{3} \pi^{3}} \int_{0}^{L}(2 x-L) \cos \frac{(2 n-1) \pi x}{2 L} d x \\ &= \frac{384 L^{3}}{(2 n-1)^{4} \pi^{4}}\left[(2 x-L) \sin \frac{(2 n-1) \pi x}{2 L}\right]_{0}^{L}-2 \int_{0}^{L} \sin \frac{(2 n-1) \pi x}{2 L} d x \\ &= \frac{384 L^{3}}{(2 n-1)^{4} \pi^{4}}\left[(-1)^{n+1} L+\frac{4 L}{(2 n-1) \pi}\right] \\ &= \frac{384 L^{3}}{(2 n-1)^{4} \pi^{4}}\left[(-1)^{n+1} L-\frac{4 L}{(2 n-1) \pi}\right] \\ &= \frac{384 L^{4}}{(2 n-1)^{4} \pi^{4}}\left[(-1)^{n}+\frac{4}{(2 n-1) \pi}\right];\end{aligned}\]

\(\lim_{x\to 0} \frac{3^x - 2^x}{x}\)

Find the surface area of the surface with parameterization \( \mathbf{r}(u,v) = \langle u+v, u^2, 2v \rangle \), \( 0 \le u \le 3 \), \( 0 \le v \le 2 \).

Endpoints of the conjugate axis located at \( (0, 3), (0, -3) \) and foci located \( (4, 0), (-4, 0) \).

Evaluate $e^{3 \ln 2}$.

\(\tan x-\sec x\)

Let \( n \) be a positive integer. Explain why \[ \binom{n}{0} + \binom{n}{1} + \binom{n}{2} + \dots + \binom{n}{n} = 2^n. \]

[T] Use a CAS to create the intersection between cylinder $9x^2 + 4y^2 = 18$ and ellipsoid $36x^2 + 16y^2 + 9z^2 = 144$, and find the equations of the intersection curves.