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The function,f: [-1,1]×[2,1] → R, defined by
f(x,y)= { x ; y is rational
{ 0 ; y is irrational
is integrable or not.
True or false with full explanation
Let f: R^3→R be function defined by :
F(x,y,z)= |x+2y+z| . Show that f is not differentiable at the point (1,-1,1).
A random sample of size n is drawn from a uniform population over (θ-1/3,θ+1/3),
Obtain maximum likelihood estimator of θ.
Graph the function "f(x)=2x+5." Use the graph to find the indicated limit, if it exists.
"\\lim\\limits_{x\\to\\infin} f(x)".
1) Find the sum of the geometric series 3 + 2 + 4 3 + 8 9 + . . .
(2) Prove that the following series converges, and find its sum 0.6 + 0.06 + 0.006 + 0.0006 + 0.00006 + · ·
Astrit will tie the goat with a 5 m long rope. One side of the rope is tied to the goat, and the other to a meadow ring that slides over an 18 m long iron rod. Calculate the surface of the lawn that the goat can reach if Astrit places the rod along the TU and UV ribs, as shown in the figure. Where should Astrit place rod b in order for the goat to reach the largest surface of the grass? Explain your answer.
Find the volume of the solid generated when the region R bounded by the given curves is revolved about the indicated axis, Sketch the Graph.
Circular Disk Method:
- R:y=√𝑥, 𝑦=4, 𝑦−𝑎𝑥𝑖𝑠 axis of revolution: 𝑦=4
- 𝑅:𝑦=𝑥2, 𝑥=3, 𝑥−𝑎𝑥𝑖𝑠 axis of revolution: 𝑥=3
Circular Ring Method:
- 𝑅: 𝑦=𝑥2, 𝑦=2𝑥 axis of revolution: 𝑥−𝑎𝑥𝑖𝑠
- 𝑅:𝑦2=4𝑥, 𝑥2=4𝑦 axis of revolution: 𝑥=4
Cylindrical Shell Method:
- 𝑅:𝑦=𝑥2, 𝑦=2𝑥 axis of revolution: 𝑥−𝑎𝑥𝑖𝑠
- 𝑅: first-quadrant region bounded by 𝑦=4𝑥 and 𝑥3=𝑦 axis of revolution: 𝑥=2
- An oil tank in the shape of a sphere has a diameter of 60 ft. How much oil does the tank contain if the depth of the oil is 25ft?
A small rectangular warehouse is to be constructed which is to have an area of 10000 square feet. The building is to be partitioning internally in to eight equal parts. The costs have been estimated based on exterior and interior walls dimensions. The costs are $200 per running foot of exterior wall and plus $100 per running foot of interior wall. [06] a) Determine the dimensions which will minimize the construction costs? b) What are the minimum costs?
y(x) =(sinx) linx
Evaluate the following indefinite integrals:
a) ∫ (1/x + 3/x2 - 4/x3 ) dx
b) ∫ (x2 + 2x - 5) / √x dx
c) ∫ x ex dx
