10 Calculus Problems

Please see the attached file for the fully formatted problems. Problems involve: parametric equation of line segment, volume of a parallelipiped, sketching a plane gven the equation, finding rectangular equations, center and radius of a sphere using the equation of a sphere, force vector problems.

Maximum of a function in a closed interval.

Find the maximum of the following function f(x)=x(1-x) over [0,1]

Derivative

Find f'(0) where f(x)=(x+1)(x+2)...(x+1000)

A formula that permits to express one class of finite sums in explicit form.

Deduce the recurrent formula for calculation of the finite sums of natural numbers in a natural power: 1^p + 2^p + 3^p + … + N^p

Cylindrical and spherical coordinates.

Write the equations i) x^2 - y^2 - 2z^2 = 4 and ii) z = x^2 - y^2 in a) cylindrical coordinates b) spherical coordinates give detailed explanation for each step of the solutions.

Equation of a Surface

Describe in words the surface whose equation is given [note r - cylindrical coordinate, ρ - spherical coordinate] a) r = 3 b) ρ = 3 c) φ = π/2 d) θ = π/3 Give detailed explanation.

parallel planes

Which of the following planes are parallel? Are any of them identical? P1: 4x - 2y - 6z = 3 P2: 4x - 2y - 2z = 6 P3: -6x + 3y -9z = 5 P4: z = 2x - y - 3 please explain each step in detail

Parallel or Perpendicular Planes

Find if the planes are parallel, perpendicular or neither. If they are not parrallel then find the equation for the line of intersection. z = x = y , 2x - 5y -z = 1 Verify that your answer is indeed a line of intersection!

angle between a cube's diagonal

Find the angle between a cube's diagonal and one of its sides. (use the vector calculus to get your answer) give detailed response. explain each step.

Orthogonality

Find a unit vector that is orthogonal to both i + j and i + k give detailed explanation for each step