Calculus Answers

let f(x)=x^4-2x^2. Find the all c (where c is the interception on the x-axis) in the interval (-2,2) such that f'(x)=0 using Rolle's theorem.

for g(x)=(x-4)/(x-3), we can use the mean value theorem on [4,6]. hence determine c

Determine whether Rolle's theorem can be applied to f on the close interval [a,b]. if it can be applied, find the values of c in open interval (a, b) such that f'(c)=0. f(x)=(x^2-2x-3)/(x+2), [-1,3]

Compute the first three derivatives of f(x)=2x^5+x^(3/5)-1/2x

A blow-moulded polymer container can be considered as a cylinder with flat ends. Its capacity is 1 litre and it has thin walls of uniform thickness.
 Produce expressions for its volume and surface area.
 Using differential calculus, find the dimensions of the cylinder which will result in the minimum amount of polymer being used for its manufacture.

Trace the curve y^2 = (x +1)(x −1)^2 by showing all the properties you use to trace it.

Apol is a company that sells smartphone. It relies on two companies: x and y to help manufacture some components in the smartphone. The cost in RM to manufacture some components in the smartphone is given by the following model,
C(x,y)=e^x (32x-16x^2-7)+2ye^x - e^2x -y^2.
Based on the mathematical model above, find the extreme points. Then, analyse the relation between the two companies and the cost to produce some components in the smartphone by Apol.

Use appropriate software to graph the surfaces
Z=7-(x^2+y^2)
and
z=1-2x+6y
on a
common screen using the domain
|x|≤10, |y|≤10
and observe the curve of intersection
of these surfaces. Show that the projection of this curve onto the xy–plane is a circle by
showing the calculation and the details of the circle equation.

A solid in the shape of a hemisphere with a radius of 2 units, has its base in the xy-plane and the centre of the base at the origin. If the density of the solid is given by the function ρ(x, y,z) = xyz, determine the mass of the hemisphere.

Show that for two scalar fields f and g:
∇.[∇f ×( f ∇g)]= 0.