Math Problem Solver

f(x, y, z) = 3x2e3y – y3e5z + 5ze–x, at the point A(1, 1, 1).

3.1

3.2

Given the point B(–120, 34, 15), write down the direction vector AB.

Obtain grad f, and calculate the value of this vector at the point A.

We can calculate the rate of change of f(x, y, z) at a point A in a general direction u.

This is known as the directional derivative, and is given by

radf

D f , u

= g

u u

where grad f is evaluated at A.

3.3

3.4

Obtain, rounding your final answer in a suitable way, the directional derivative

of f(x, y, z) at A in the direction AB.

Verify that grad f in 3.2 is irrotational.

Consider the double integral

15xydxdy

where R is the triangular region in the first

quadrant bounded by the y-axis, and the lines y = 5x and x + 2y = 33.

3.5

3.6

Sketch the region R.

By first setting up suitable limits, evaluate this double integral, correct to two

decimal places