Calculus Answers

Find the work done in stretching a spring of natural length 8 cm, from 10 cm to 13 cm. Assume a force of 6 N is needed to hold it at a length of 11 cm.

check whether the function f (x,y)={4x^2y/4x^4+y^2 , (x,y)not=to (0,0) 0 (x,y)= to (0,0). } Is continuous at (0,0) ? Justify your answer.

Find the center of mass (gravity) of a solid of constant density that is bounded by the parabolic
cylinder x = y
2
and the planes x = z, z = 0, and x = 1. (

a) integrate (3x+1) √4x^2+12x+5. dx
b) integrate x^2+x+5/(x^2+4)(x+1) dx

a) integrate (3x+1)√4x^2 + 12x + 5 dx
b) integrate x^2+x+5 / (x^2+4)(x+1)

Find the absolute maximum and minimum values of the following function over the given interval. If there are multiple points in a single category list the points in increasing order in x value and enter N in any blank that you don't need to use.

Cersei is the ruler of a country and her army is fighting a war. The probability of her side winning
the war depends upon two variables - the average fitness of her army (given by x) and the morale of
her army (denoted by m). Furthermore, both the fitness and morale are functions of how much food
is available to her army. If f kilogram of food is available to her army (per person per day), then
x(f) = −f
2 + 5f and m(f) = f − 1 (where f ∈ [0, 5]). The probability of her side winning the
war is given by a function of x and m and is given by the function g such that g(x, m) = x
2+2m+2
40 .
Currently, her army is getting 1.5 kg of food per person per day. By approximately how much will
her army’s probability of winning change if she increases the supply of food by a very small amount?

Compute fxy and fyx for the function f(x,y) = e^(x+y) +9x^2 +2xy at (1,2).

Check whether the function
f(x,y)= { 4x^2y/4x^4+y^2,. (x,y)not=to(0,0)
0 (x,y) = to (0,0)}

is continuous at ( 0,0).