Problem 19 : Order of the differential equation

Find the order of the differential equation of all parabola with axis parallel to the axis of y. ( DC/M)

SOLUTION OF PROBLEM: 12

Find the values of a, b, c and d for the given limit.  (DC/M)
SOLUTION:

SOLUTION OF PROBLEM: 10

Let function f is from R to R be defined by f(x) =2x + sin(x) for x belongs to R. Then f is

a) one-to-one and onto

b) one-to-one but not onto.

c) onto but not one-to-one

d) neither one-to-one nor onto.      ( DC/CH)

SOLUTION: The given function has co-domain as R( set of all real numbers)

For the function f(x) = 2x + sin(x) , 2x ranges from negative infinity to positive infinity whereas sin(x) varies from [-1,1] so ultimately the f(x) will have its range as R. Therefore, the f(x) is Onto function.

Also, f'(x) = 2 + cos(x) , cos(x) will vary from [-1,1], So 2+cos(x) varies from [1,3].

Therefore, f'(x) is always positive. Hence f(x) will always be an increasing function.Therefore, f(x) will be one-to-one.

So, the correct option is (a)

SOLUTION OF PROBLEM: 8

Prove that the f(x) = 2(x)cube + 3(x)sq+6x +10 will always be an increasing function. (DC/M)

Solution: If we differentiate the given function we will get f'(x) = 6(x)sq + 6x +6 = 6( (x)sq +x +1)

Now, we can write f'(x) = 6[(x + 1/2)sq +3/4] which is always positive.So f(x) will always be an increasing function.

Alternatively, we can also say the for the quadratic expression (x)sq +x +1 coefficient of (x)sq is positive and discriminant is negative (-3), so the expression is always positive

INCREASING FUNCTION

PROBLEM: 8

Prove that the f(x) = 2(x)cube +3(x)square +6x +10 will always be an increasing function. (DC/M)

OUR SOLUTION OF PROBLEM: 2

Find the interval in which 
f(x)=2log(x-2)-xsq+4x+1 increases
a) (1,2)
b) (2,3)
c) (5/2,3)
d) (2,4)

Note: xsq term stands for x square.                         (DC/M)

SOLUTION:

Before starting up the problem, let’s first of all find out the domain of the given function

Since, we have (x-2) is inside the log it should be positive, it will happen only when x>2.

Hence the domain of given function is x > 2 therefore option a) is ruled out.

Now differentiate the given function, we get f ‘(x)= -(x-3)(x-1)/ (x-2)

For f(x) to increasing, f ‘ (x) > 0

Since, x> 2, the denominator of f ‘ (x ) is always positive, which means for f (x) to increasing -(x-3)(x-1) >0

which means x will lie from ( 1,3), but since the function is defined for x >2, we will have (2,3 )as the interval where function is increasing.

Hence the correct choices are (2,3) and (2.5 ,3)

 

DIFFERENTIAL CALCULUS

PROBLEM: 2

Find the interval in which f(x)=2log(x-2)-xsq+4x+1 increases
a) (1,2)
b) (2,3)
c) (5/2,3)
d) (2,4)                                                   ( DC/M)


Note: xsq term stands for x square.