Thursday, 17 September 2026

Show that there are no prime triplets other than 3,5,7.

Use the multiplication of power series to verify that $e^x e^{-x} = 1$.

Solve the initial value problem \[ y''' - 6y'' + 11y' - 6y = 0, \quad y(0) = 4, \quad y'(0) = 5, \quad y''(0) = 9. \]

Find a particular solution of $$y^{\prime\prime}-3 y^{\prime}+2 y=e^{-2 x}[2 \cos 3 x-(34-150 x) \sin 3 x]$$

\(Finding the surface area of a parametrized surface Using the parametrization found in Example 14.5.2, find the surface area of z = x^2 + 2y^2 over the circular disk of radius 2, centered at the origin.\)

Converting from rectangular to parametric Consider $y = x^2$. Find parametric equations $x = f(t), y = g(t)$ for the parabola where $t = \frac{dy}{dx}$. That is, $t = a$ corresponds to the point on the graph whose tangent line has slope $a$.

\(r = 2\theta\)

$$x=t^2, \quad y=2 \ln t, \quad t \ge 1$$

Let $n$ and $k$ be integers with $0 \le k < n$. Explain why \[ \binom{n}{k+1} = \binom{k}{k} + \binom{k+1}{k} + \dots + \binom{n-1}{k}. \]

Suppose vectors \(\mathbf{u}\) and \(\mathbf{v}\) lie in the xy-plane (the z-component of each vector is zero). Now suppose the x- and y-components of \(\mathbf{u}\) and the y-component of \(\mathbf{v}\) are all positive, whereas the x-component of \(\mathbf{v}\) is negative. Assuming the coordinate axes are oriented in the usual positions, in which direction does \(\mathbf{u} \times \mathbf{v}\) point?