Calculus Answers

1 Find the angle between
A=2x+2j−k
A=2x+2j−k
and
B=6i−3j+2k
2 Determine the value of a so that
A=2i+aj+k
A=2i+aj+k
and
B=4i−2j−2k
B=4i−2j−2k
are perpendicular
3 Determine a unit vector perpendicular to the plane of
A=2i−6j−3k
A=2i−6j−3k
and
B=4i+3j−k
4 Find the work done in moving an object along a vector
r=3i+2j−5k
5 Given that
A=2i−j+3k
A=2i−j+3k
and
B=3i+2j−k
B=3i+2j−k

, find
A⋅B
6 If
A=2i−3j−k
A=2i−3j−k
and
B=i+4j−2k
B=i+4j−2k
, find
(A+B+×(A−B)
7 If
A=3i−j+2k
A=3i−j+2k
,
B=2i+j−k
B=2i+j−k
and
C=i−2j+2k
C=i−2j+2k
, find
(A×B)×C
8 Determine a unit vector perpendicular to the plane of
A=2i−6j−3k
A=2i−6j−3k
and
B=4i+3j−k
9 Evaluate
(2i−3j)⋅[(i+j−k)×(3i−k)]
10 If
A=i−2j−3k
A=i−2j−3k
,
B=2i+j−k
B=2i+j−k
and
C=i+3j−2k
C=i+3j−2k

,evaluate
(A×B)⋅C

7 If
A=5t2+tj−t3k
A=5t2+tj−t3k
and
B=sinti−costj
B=sin⁡ti−cos⁡tj
. evaluate
ddt(A⋅A)
8 If
A=sinui+cosuj+uk
A=sin⁡ui+cos⁡uj+uk
,
B=cosui−sinuj−3k
B=cosui−sinuj−3k
and
C=2i+3j−k
C=2i+3j−k
, evaluate
ddu(A×(B×C))
ddu(A×(B×C))

at u=0
9 Let
A=x2yzi−2xz3j−xz2
A=x2yzi−2xz3j−xz2
and
B=4zi+yj+4x2k
B=4zi+yj+4x2k
, find
∂2∂x∂y(A×B)
∂2∂x∂y(A×B)

at (1,0,-2)

10 solve
d2Adt2−4dAdt−5A=0

1 Given that
A=sinti+costj+tk
A=sin⁡ti+cos⁡tj+tk

, evaluate
∣∣∣d2Adt2∣∣∣
2 A particle moves along a curve whose parameter equations are
x=e−t
x=e−t
,
y=2cos3t
y=2cos⁡3t
,
z=2sin3t
z=2sin⁡3t
. Find the magnitude of the acceleration at t=0
3 A particle moves along the curve
x=2t2
x=2t2
,
y=t2−4t
y=t2−4t
and
z=3t−5
z=3t−5
,where t is the time. Find the components of the velocity at t=1 in the direction
i−3j+2k
4 Determine the unit tangent at the point where t=2 on the curve
x=t2+1
x=t2+1
,
y=4t−3
y=4t−3
and
z=2t2−6t
5 If
A=5t2+tj−t3k
A=5t2+tj−t3k
and
B=sinti−costj
B=sin⁡ti−cos⁡tj

. evaluate
ddt(A⋅B)

Suppose the rate of savings in Malaysia is given by

dS/dt=5.1e^t+400t^2+6000


where t is the time in years, and S is the amount of money saved in billions of RM. Find the function S(t) which gives the total amount of money saved. You may assume that RM0 is saved at t = 0.

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?

A manufacturer of furniture has determined that the weekly production function for p thousand furniture is

p=1000+20l^2 k^3-5l^3-3k^4

where l is the number of labor hours per week in thousands and k is the amount of capital in thousands of RM per week. Determine:

i) the marginal production function with respect to l

ii) the marginal production function with respect to k

Q. Expand the expression ∑_i∑_j∑_k▒aikbijxkxj

lim Fn+1/Fn = e (euler's number)
n->infinity

Fn is a Fibonacci Number

Please I need to know ;-;

evaluate the following using Beta function:

I=integration(dx/sqrt(1-(x)^n)) limits: from zero to one

thanks in advance.

evaluate the following using Beta function:

I=integration(dx/sqrt(1-(x)^n)) limits: from zero to one

thanks in advance.