Calculus Answers

An airplane at an altitude at an altitude of 4400 feet is flying horizontally away from an observer. At the instant when the angle of elevation is 45 degrees, the angle is decreasing at the rate of 0.5 rad/sec. How fast is the airplane flying at that instant?

A closed box with a square base is to have a volume of 2000 cu inches. The material for the top and bottom of the box is to cost 3 pesos per square inch, and the material for the sides is to cost 1.50 pesos/square inch. If the cost of the material is to be least, find the dimensions of the box.

Find the numbers whose sum is x , if the product of one by the square of the other is to be a maximum.

A particle moves along a straight line according to the law : S = 132 + 10t - 6t^2 + 3t^3.
Find the following:
a. The velocity and acceleration at any time t?
b. Velocity at t = 2;
c. Acceleration at t = 3.

Find the equation of the line tangent to the curve y = 3x^2 - 4x and parallel to the line
x - 2y + 6 = 0

Find the tangent line to the curve y = x^4 + 4x^3 - 8x^2 + 3x + 70 with slope 3.

Find the tangent line to the curve y = x^4 + 2x^3 - 2x^2 - 3x + 3 perpendicular to the line x - 3y = 2

Antiderivative of (sec^2(4)x+1)dx

(b) To fill an order for 100 units of a product, a firm wishes to distribute the production between its two plants, Plant 1 and Plant 2. The total cost function is given by

c=f(q_1,q_2 )=0.5q_1^2+2q_1+32q_2+500


where q_1 and q_2 are the number of units produced at Plants 1 and 2, respectively. How should the output be distributed in order to minimize costs?

1 If
ϕ=2xz4−x2y
ϕ=2xz4−x2y
, find
|▽ϕ|

2 If
ϕ(x,y,z)=3x2y−y3z2
ϕ(x,y,z)=3x2y−y3z2
, find
▽ϕ
▽ϕ
at point (1,-2,-1)

3 Find a unit normal to the surface
x2y+2xz=4
x2y+2xz=4
at point (2,-2,3)
4 Let
ϕ(x,y,z)=xy2z
ϕ(x,y,z)=xy2z
and
A=xzi−xy2j+yz2k
A=xzi−xy2j+yz2k

,find
∂3∂x2∂z(ϕA)
5 Given that
ϕ=2x2y−xz3
ϕ=2x2y−xz3

find
▽2ϕ
6 If
A=xz3i−2x2yzj+2yz4
A=xz3i−2x2yzj+2yz4
, find
▽×A
▽×A

at point (1,-1,1).

7 Given that
A=A1i+A2j+A3k
A=A1i+A2j+A3k
and
r=xi+yj+zk
r=xi+yj+zk
, evaluate
▽⋅(A×r)
▽⋅(A×r)

if
▽×A=0
8 Let A=x2yi−2xzj+2yzk
A=x2yi−2xzj+2yzk

, find Curl curl A.
9 Given A=2x2i−3yzj+xz2k
A=2x2i−3yzj+xz2k
and
ϕ=2z−x3y
ϕ=2z−x3y
, find

10 Find the directional derivative of
ϕ=x2yz+4xz2
ϕ=x2yz+4xz2
at (1,-2,-1) in the direction
2i−j−2k
A⋅▽ϕ
A⋅▽ϕ
at point (1,-1,1)