Questions from General Calculus

Q: (a). Show that a differentiable function f decreases most rapidly

(a). Show that a differentiable function f decreases most rapidly at x in the direction opposite to the gradient vector, that is, in the direction of 2=f (x). (b). Use the result of part (a) to find t...

Q: Find all points at which the direction of fastest change of the

Find all points at which the direction of fastest change of the function f (x, y) = x2 + y2 - 2x - 4y is i + j.

Q: Near a buoy, the depth of a lake at the point

Near a buoy, the depth of a lake at the point with coordinates (x, y) is z = 200 + 0.02x2 - 0.001y3, where x, y, and z are measured in meters. A fisherman in a small boat starts at the point s80, 60d...

Q: A cardboard box without a lid is to have a volume of

A cardboard box without a lid is to have a volume of 32,000 cm3. Find the dimensions that minimize the amount of cardboard used.

Q: A rectangular building is being designed to minimize heat loss. The

A rectangular building is being designed to minimize heat loss. The east and west walls lose heat at a rate of 10 units/m2 per day, the north and south walls at a rate of 8 units/m2 per day, the floor...

Q: If the length of the diagonal of a rectangular box must be

If the length of the diagonal of a rectangular box must be L, what is the largest possible volume?

Q: A model for the yield Y of an agricultural crop as a

A model for the yield Y of an agricultural crop as a function of the nitrogen level N and phosphorus level P in the soil (measured in appropriate units) is Y (N, P) = kNPe-N-P where k is a positive...

Q: The Shannon index (sometimes called the Shannon-Wiener index or

The Shannon index (sometimes called the Shannon-Wiener index or Shannon-Weaver index) is a measure of diversity in an ecosystem. For the case of three species, it is defined as H = -p1 ln p1 - p2 ln...

Q: Three alleles (alternative versions of a gene) A, B

Three alleles (alternative versions of a gene) A, B, and O determine the four blood types A (AA or AO), B (BB or BO), O (OO), and AB. The Hardy-Weinberg Law states that the proportion of individuals i...

Q: Where does the normal line to the paraboloid z = x2 +

Where does the normal line to the paraboloid z = x2 + y2 at the point (1, 1, 2) intersect the paraboloid a second time?