Calculus Answers

Find the two repeated limits of the function f(x ,y) = (y-x/y+x) (1+x^2)/1+y^2) at (0,0) . Does
the simultaneous limit of f exist as (x, y) →(0,0) ? Give reasons for your answer.

Check whether the limit of the function
f(x,y)=3x^3y/x^6+2y^2 exists as (x, y) →(0,0)

State whether the following statements are true or false. Give reasons for your answers.
a) The function R^3 → R. f : , given by f (x, y,z) = |x| + |y| + |z| is differentiable at
3,2(-1)

b) The function
f(x,y)=max{y/x,x} is a homogeneous function on R^2

c) The domain of the function f / g where f (x, y) = 2xy and g(x, y) = x^2 + y^2 is R^2 .

a) Obtain the 6th roots of (-7) , represent them in an Argand diagram.
b) using epsilon Delta definition , show that Lim(3x-5)=1
x->2

Which of the following statement are true and which are false ? Give reason for your answers in the form of a short proof or a counterexample .
a) there are at least two ways of describing the set {7,8....}
b) any function with domain R X R is a binary operation.
c) the graph of every function from [0,1] to R is infinite .
d) The function f: R --> R , defined by f(x)= x|x| , is an odd function.
e) the domain of the function f ° g ,where f(x) = √x and g(x) = √ 2-x , is [- infinity,2]

The displacement, y(m), of a body is damped oscillation is y=2e^-t sin3t.

use product rule to find an equation for the velocty of object if v=dy/dt.

The displacement of a mass is given by the function y=sin3t
The tasks are to:
1. Draw a graph of the displacement y(m) against time(s) for the time t= 0s to t= 2s.
2. Identify the position of any turning points amd whether they’re maxima, minima or points of inflexion.
3. Calculate the turning points of the function using differential calculus and show which are maxima, minima or points of inflexion by using the second derivative.

The gain of an amplifier is found to be G=20log(10Vout):
The tasks are to find equations for :
A. dG/dVout
B. d^2G/dVout^2

a communication signal is given by the function y=sint/t
derive and equation for dy/dt using the quotient rule.

Show the COMPLETE SOLUTION for the given problem.

1. Use Lagrange Multiplier to determine the dimensions of a rectangular box, open at the top, having a volume of 32 cubic feet and requiring the least amount of material for its construction.