Calculus Answers

determine the integral of function ( x^3 -2 )(x^4 -8x)^0.5 dx

Question:

A particle moves such that its vector is given by 𝑟

= cos 𝜔𝑡𝑖 + sin 𝜔𝑡𝑗 where 𝜔 is a constant. Show that:

(i)

Velocity, 𝑣

of the particle is perpendicular to 𝑟

(ii)

Acceleration, 𝑎

is directed towards the origin and has magnitude proportional to the

distance from the origin.

(iii) 𝑟

𝑋 𝑣

is a constant vector

Question:

If 𝑎 and 𝑏 are two sides of a right angled hypotenuse 𝑐 and let 𝑝 be the perpendicular from the opposite

vertex on the hypotenuse, show that:

1. (𝛿 𝑝/𝛿𝑎)_b =b³/c³

2. (𝛿𝑝/𝛿𝑎 )_A=b/c

Consider the function f(x) = 5 ; for x < 0 0, for x = 0 5, for x > 0

Show that f has no limit as x approaches 0

Question:

If 𝑎 and 𝑏 are two sides of a right angled hypotenuse 𝑐 and let 𝑝 be the perpendicular from the opposite

vertex on the hypotenuse, show that:

1. (𝛿 𝑝/𝛿𝑎)_b =b³/c³

2. (𝛿𝑝/𝛿𝑎 )_A=b/c

Question:

Find the equation of the tangent plane and normal to the surface

2𝑥𝑧² − 3𝑥𝑦 − 4𝑥 = 7 at point

( 1, −1,2 )

Question 2

(a) Given that A ~ = (x + 2y + 4z)ˆ

i + (2x - 3y- z)ˆj +(4x-y + 2z)kˆ

(i). Show that the vector field A ~ is irrotational

(ii). Find the scalar potential φ such that A~ = ∇φ, if φ(0, 0, 0) = 1

Show that ∇𝑟ⁿ = 𝑛𝑟^n-2"\\vec r"