Directional derivative

Find the directional derivative of f at P in the direction of v; that is find D_u f(P), where u=v/{v}: f(x, y, z)= ln(1 + x^2 +y^2 - z^2) ; P(1, -1, 1), v=2i - 2j -3k

Maximum directional derivative

Find the maximum directional derivative of f at P and the direction in which it occurs: f(x, y)= sin (3x - 4y) ; P(pi/3, pi/4)

Normal gradient vector

Use the normal gradient vector to write an equation of the line (or plane) tangent to the given curve (or surface) at the given point P: x^(1/3) + y^(1/3) + z^(1/3) = 1; P(1, -1, 1)

Rate of change

Suppose that the temperature at the point (x, y, z) in space (in degrees Celsius) is given by the formula: W= 100 - x^2 - y^2 - z^2. The units in space are meters. (a) Find the rate of change of temperature at the point P(3, -4, 5) in the direction of the vector v=3i - 4j + 12k. (b) In what direction does W increase most rapi ...continues

Logarithms

Please see the attached file for full problem description.

Find the common logarithm of

Please see the attached file for full problem description.

Vectors

Please see attachment. Require problems solving, also explanations etc for better understanding of vectors. VECTOR PROBLEMS (1) Let l be the line with equation v = a + t u. Show that the shortest distance from the origin to l can be written | a × u | ...continues

LAPLACE TRANSFORMS

Please see attachment. Require problems solving, also explanations etc for better understanding.

Prime numbers

Prove that if p is a prime number, then p divides , for all n≥p. Here [r] denotes the greatest integer ≤ r , for any real number r. Does this result generalize to a result about instead of p ?

Induction

Prove by induction where n is a positive integer. (The questions are attached).