Calculus Answers

3. a) Define tangent and normal of a curve with figure. Also find the equation of tangent and normal of the ellipse (x ^ 2)/4 + (y ^ 2)/16 = 1 at the point (- 1, 3) .

b) Explain maximum and minimum value of a function with graphically. Evaluate maximum and minimum value of the function f(x) = x ^ 3 - 3x ^ 2 + 3x + 1

∫ ௫ିଵ ௫ యି௫మିଶ௫ 𝑑𝑥 

1.       Find the function whose tangent has slope 4x + 1 for each value of x and whose graph passes through the point (1, 2).

Given that U

 is a function of x,y

 and z

 and A

 a vector field, prove that:

∇×(UA)=(∇U)×A+U(∇×A).

a) Define Bijective function and Surjective function.

b) Find the limiting value of i) lim

𝑥→7

𝑥

2+2𝑥−63

𝑥−7

ii) lim

𝑥→∞

5𝑥−1

5𝑥+1

.

c) Test the continuity of following functions at x= -2 and x=3

𝑓(𝑥) = {

7𝑥 − 1 𝑖𝑓 𝑥 > 3

𝑥

2 − 8 𝑖𝑓 − 2 ≤ 𝑥 ≤ 3

8𝑥 + 3 𝑖𝑓 𝑥 < −2

If A and Bare vector fields, prove the following:

nabla(A* B)=(B* nabla)A+(A* nabla)B+B*( nabla* A)+A*( nabla* B) .

[SADT10] Let r=x hat i +y hat j +z hat k and r = ||r||

Show that:

nabla(lnr)= r r^ 2 .

and

nabla*(r^ n r)=0 .

[SADT9] The Laplacian of a function f of n variables x 1 ,x 2 ,*** x n denoted nabla^ 2 f is defined by

nabla^ 2 f(x 1 ,x 2 ,***,x n ):= partial^ 2 f partial x 1 ^ 2 + partial^ 2 f partial x 2 ^ 2 +***+ partial^ 2 f partial x n ^ 2

Now assume that f depends only on r where r=(x 1 ^ 2 +x 2 ^ 2 +***+x n ^ 2 )^ 1 2 i.e. f(x 1 ,x 2 ,***,x n )=g(r) for some function g. Show that, for x 1 ,x 2 ,***,x n ne0 ,

nabla^ 2 f= n-1 r g^ prime (r)+g^ prime prime (r)