List of top Signals and Systems Questions on Zener Diodes

Consider a vector $ \Bar{u} = 2\hat{x} + \hat{y} + 2\hat{z} $, where $ \hat{x}, \hat{y}, \hat{z} $ represent unit vectors along the coordinate axes $ x, y, z $ respectively. The directional derivative of the function $ f(x, y, z) = 2 \ln(xy) + \ln(yz) + 3 \ln(xz) $ at the point $ (x, y, z) = (1, 1, 1) $ in the direction of $ \mathbf{u} $ is:

The input $ x(t) $ and the output $ y(t) $ of a system are related as \[ y(t) = e^{-t} \int_{-\infty}^t e^t x(\tau) d\tau, \quad -\inftyLt;tLt;\infty. \] The system is:

Consider the discrete-time systems $ T_1 $ and $ T_2 $ defined as follows:
$ [T_1x][n] = x[0] + x[1] + \dots + x[n], $
$ [T_2x][n] = x[0] + \frac{1}{2}x[1] + \dots + \frac{1}{2^n}x[n]. $
Which of the following statements is true?

If the Z-transform of a finite-duration discrete-time signal $ x[n] $ is $ X(z) $, then the Z-transform of the signal $ y[n] = x[2n] $ is: