Calculus Answers

1. The profit of a company is given by P(x) = 5 000 +1 000x-5x² where x is the amount (in thousands of pesos) that the company spends on advertising.

a. Find the amount x that the company has to spend to maximize its profit.

b. Find the maximum profit.

Can You Show Me the Way? Complete the table of values of the exponential function. Then, draw its graph in the given coordinate plane. 2 3 1 х -3 -2 -1 0 f(x) = 3*1. How will you describe the graph? 2. In which quadrant/s do the graph occupies?3. What are the other possible values of x? Describe the range of the possible values of x? 4.What do you think now is the range of the function? 5. What other observations can you give?

2. Use change of variables technique to integrate the following

Z

x(3x − 2) 1

2 dx

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).