Calculus Answers

company can produce and sell x-units of one commodity and y-units of other commodity

at a pro€t given as P(x, y) = 400x + 500y − x

2 − y

2 − xy − 20000. Find the units and amount

for which the pro€t is maximum and minimum.

‘e lengths p, q, and r of the edges of a rectangular box are changing with time. At the

instant p = 2m, q = 3m, r = 4m,

dp

dt =

dq

dt = 1 m/sec and dr

dt = −2 m/sec. At what rate is the

box’s volume V changing at that instant?

A cellular phone company has the following production function for a smart phone: p(x, y) =

50x

2

3 y

1

3 where p is the number of units produced with x units of labor and y units of capital. a)

Find the number of units produced with 125 units of labor and 64 units of capital. b) Find the

marginal productivities (Hints: Partial derivatives). c) Evaluate the marginal productivities at

x = 125 and y = 64.

For the function v = 12 sin 40, calculate the: a) mean b) root mean square (RMS) 

Over a range of 0 ≤ Ø ≤ π/4 radians. 

(Note: the trigonometric identity cos 2Ø = 1-2sin^2 Ø)

Determine whether the following series converge, converge absolutely, converge conditionally, or diverge.

1. "\\displaystyle\\sum_{k=1}^\t\u221e" "(\u221ak+1\/\u221ak)"^-3
2. "\\displaystyle\\sum_{k=1}^\t\u221e" "(cos(k\\pi\/2)\/(2k)\\pi"
3. "\\displaystyle\\sum_{k=1}^\t\u221e"(-1)^k+1 "(k-1)!(k+1)!\/(2k)!"
4. "\\displaystyle\\sum_{k=1}^\t\u221e" "(1.3.5...(2k-1))\/(1.4.4.7...(3k-2)))"

Find the equation of the normal line to the curve of the equation x 2 y + xy2 = 6 at the point (2, 1).

Using the Intermediate Value Theorem and a calculator, find an interval of length 0.01 that contains a root of e^x = 2 - x, rounding interval endpoints off to the nearest hundredth.

Xy^2z^3=8 at (2,2,1)

find the equation of the tangent plane and the normal line given the surface xy^2z^3=8 at (2,2,1)