Consider the surface f(x, y) = 21 - 4x² - 16y² (a plane) and the point P(1,1,1) on the surface. Required:
a. Find the gradient of f.
b. Let C' be the path of steepest descent on the surface beginning at P, and let C be the projection of C' on the xy-plane. Find an equation of C in the xy-plane.
c. Find parametric equations for the path C' on the surface.

Colby and jaquan are growing bacteria in an experiment in a laboratory. colby starts with 50 bacteria in his culture and the number of bacteria doubles every 2 hours. jaquan starts with 80 of a different type of bacteria that doubles every 3 hours. let x equal number of days. colbys experiment follows the model: a.) y=50*2^x b.) y=50*2^8x c.) y=50*2^12x jaquans experiment follows the model: a.)80*2^x b.)80*2^8x c.)80*2^12x

Describe the domain and range of y=2(3)^x-2+5

The focus of parabola is (-4, -5), and its directrix is y= -1. fill in the missing terms and signs in parabolas equation in standard form