Calculus Answers

Consider a lamina that occupies the region D bounded by the parabola x = 1 - y² and the coordinate axes in the first quadrant with density function p (x, y) = y. Find the center of mass.

Determine the surface area of the portion of 2x + 3y + 6z = 9 that is in the 1st octant.

Define h(0,0) in such a way that h is continuous at origin h(u,v)=ln⁡((nu^2-u^2 v^2+nv^2)/(u^2+v^2 ))

Is the function h(u,v)=cos⁡(u)+e^nu+√(sinh⁡(u^2 )+√(v+e^nu ))  , Where n is second digit of your arid number for example if your arid number is 19-arid-12345 then choose n=2. Is is continuous? In case of yes then find the points of continuity. Explain briefly

Find the partial derivative at point (u,v)=(-4,0) of w=ln⁡(x^α+y^β+z^α ),x=ue^v sinu,y=ue^v cosu,z=ue^v, 

where α and β are largest and second largest digits of your arid number e.g if your arid number is 20-arid- 470, then α= 7, β=4.

Use the limit definition to find partial derivative of

 f(u,v,w)=u^2-nuv+(n+1) v^2-(n+2)uw+(n+3)vw^2-(n+4)u+(n+5)v-(n+6)w,Where n is the sum of your arid number e.g if 20-arid-470 choose n=4+7+0=11. 

. Consider a function h(u,v)=4u^2+v^2. Find the level curves for k=n+1,n+2,n+3,n+4,n+5,n+6,where n is the sum of second and third digit of your arid number e.g.20-arid-470 take n=7+0=7. Perform all steps clearly. Also Sketch the neat Graph 

. Use Numerical Method to find Limit of function h(u,v)=(u^4-v^4)/(u^4+v^4 ) . Find lim┬((u,v)→(n,n))⁡〖h(u,v).〗 Where n is your arid number for example if your arid number is 19-arid-12345 then choose n=12345. Choose at least 4 most nearest values of n for both (u,v)(Don’t choose far values from n marks will be deducted in that case). Construct neat table and also perform all calculations. And check whether the limit exist or not? If yes then what’s the value of limit. 

 Define h(0,0) in such a way that h is continuous at origin h(u,v)=ln⁡((nu^2-u^2 v^2+nv^2)/(u^2+v^2 ))

Define h(0,0) in such a way that h is continuous at origin h(u,v)=ln⁡((nu^2-u^2 v^2+nv^2)/(u^2+v^2 ))