Calculus Answers

Determine the length of the arc (in radian measure) and the measure of the angle (in radian degree measures) generated by a point that starts (1,0)and terminates at the following:

1. Positive x-axis

2. Negative x-axis

3.Positive y-axis

4.Negative y-axis

Given that U is a function of x, y, and z

and A a vector field, prove that:

∇×(UA)=(∇U)×A+U(∇×A).

(a) Evaluate∫[

𝒙/(𝒙^2+𝟏)^(1/2)𝒅𝒙.

(b) Use MATLAB to generate some typical integral curves of 𝑓(𝑥) =

𝒙/(𝒙^2+𝟏)^(1/2)𝒅𝒙over the interval (−5,5).

The Laplacian of a function f of n variables x₁, x₂,⋯x_n, denoted ∇²f is defined by ∇²f(x1, x2,⋯xn) := (∂²f/∂x₁²)+(∂²f/∂x₂²)+...+(∂²f/∂x_n²)

Now assume that f depends only on r where r= (x₁²₊ x²₂₊⋯+x²_n)^1/2

i.e. f(x₁,x₂,⋯,x_n)=g(r), for some function g

. Show that, for x₁,x₂,⋯,x_n≠0, ∇²f=[(n-1)/r]g′(r)+g′′(r)

The acceleration of an object moving in a strange way has been modelled as a = e^xx .

a)  Use integration by parts to find an equation to model the velocity  if v = ∫ = e^xx dx

b) Is the problem any different if you find v = ∫ = xe^x dx

 Let r=xi^+yj^+zk^ and r=||r||. Show that: ∇(lnr)=r/r^2 .

and

∇×(r^n r)=0.

Given that U is a function of x, y and z and A a vector field, prove that:

∇⋅(UA)=(∇U)⋅A+U(∇⋅A).

If A and B are vector fields, prove the following:

∇.(A×B)=B⋅(∇×A)−A⋅(∇×B).

Evaluate F(x)=∫ √t

2 + 9

x

4

.

a) F(4) b) F’(4) c) F’’(4)

Evaluate

∫ √tanx. sec4x dx

π

4

Also integrate the definite integral using MATLAB command.