Calculus Answers

Which of the following statements are true? Give reasons for your answers, in the form of a short proof or a counter example. (1) d^2/dx^2=(dy/dx) ^2 (2) The inverse function of y=e^3x is y=1/3lnx. (3) If f is increasing and f(x) >0 on an interval I, then g(x) = 1/f(x) is deceasing on I. (4) An equation of the tangent line to the parabola y=x^2 at (-2, 4) is y-4=2x(x+2). (5) If f is one - one onto and differentiable on R, then (f^-1) '(6) =1/f'(6)

The projection π3 defined on a domain R^3 is a vector-valued function. True or False?

Find dz/dt for z=x^2 y+4y^2 where x=cost and y=sint using the chain rule

Compute fxy and fyx for the function f (x,y)=e^x+y sinx +9x^2+2y at (1,2)

(1) Verify Euler’s relation for the function f(x,y)=(x^3+y^3)/(x+y).
(2) Check whether the following functions are differentiable at the point given
against them:
i) f(x,y)=y^3+y Sin2x+e^(x+y) at (1,–1)
ii) f(x,y)=|x=1| at (1,0)
(3) Find the range of the function f defined by f(x,y)=10–x^2–y^2 for all (x,y)
for which x^2+y^2 ≤9. Sketch two of its level curve

Find fx (0,0) and fx (x,y),where (x,y)≠(0,0) for the following function
f (x,y)=xy^3/x^2+y^2, (x,y)≠(0,0)
0, (x,y)=(0,0)
is continuous at (0,0)? Justify your answer.

Check whether the function
f (x,y)=4x^2 y/4x^4+y^2, (x,y)≠(0,0)
0, (x,y)=(0,0)
is continuous at (0,0)

(1) Calculate the Jacobian of ∂(x,y,z)/∂(u,v,w) for x=√w ucosv,y=√w usinv and z=w–1 at the point (5,π/2 ,3)
(2) Find the minimum value of the function f(x,y)=x^2+2y^2 on circle x2+y^2=1
(3) If u=Sin^(–1) (x^2+u^2)^(1/5) then show that x∂u/∂x +y∂u/∂y=(2/5)tanu
(4) Let the function f be defined by
f(x,y)=3x^2y^4/x^4+y^8 ,(x,y)≠(0,0)
=0 (x,y)=(0,0)
Show that f has directional derivatives in
all direction at (0,0)
(5) Verify that f(x,y) =exp(–k^2 t)Sin(Kx)
satisfies the heat equation ∂f/∂t=∂^2f/∂x^2 where k is a constant

Which of the following statements are true? Give reasons for your answers, in the form of a short proof or a counter example.
(1) d^2/dx^2=(dy/dx) ^2
(2) The inverse function of y=e^3x is y=1/3lnx.
(3) If f is increasing and f(x) >0 on an interval I, then g(x) = 1/f(x) is deceasing on I.
(4) An equation of the tangent line to the parabola y=x^2 at (-2, 4) is you=2x(x+2).
(5) If f is one - one onto and differentiable on R, then (f^-1) '(6) =1/f'(6) .

State whether the following statements are true or false. Justify yourself with the help of a short proof or a counter example.
(1) There are at least two ways of describing the set {7, 8....}.
(2) Any function with domain R®R is a binary operation.
(3) The graph of every function from [0, 1] to R is infinite.
(4) The function f:R®R, defined by f(x) =x|x|, is an odd function.
(5) The domain of the function f(g(x)), where f(x) =√x and g(x) =√2-x, is [-infinite,2.