Further Calculus 1 Solutions

 

 

 

1.         y = 2       =  (2x – 2x-1 – 2x-2)2

           

           

            = 2(2 + 2x-2 + 4x-3)

 

 

 

 

2.                   y  =   =  (cos(x))1/2

 

 

  =  [cos(x)]-1/2 sin(x)

 

 

            = 

 

 

 

 

3.                   y  =  1 – 2sin2(x)

 

 

  =  -4sin(x)cos(x)  =  -2sin(2x)

 

 

 

 

4.                   y = cos2(x)

 

 

  =  2cos(x)sin(x)      =          sin(2x)

 

 

at           y = cos2()   =   ()2   = 3/4  point is  (, 3/4)

 

To find the equation of a straight line require the gradient and a point.

 

 

 () = sin(2x) = sin()  =

 

 

Equation of tangent:

 

                        y – b  = m(x – a)

 

                        y – 3/4  = (x - )

 

 

                        12y - 6x = 9 -2

 

 

 

 

5.                   f(x) = 2cos(x/2) + x

 

 

 = -sin(x/2) + 1

 

The maximum value of sin(x/2) is +1 so the minimum value of -sin(x/2) is –1

 

This means that the minimum value of  is 0

 

i.e.                    since  is never negative the curve is never decreasing.

 

 

S.P’s occur  when = 0

 

i.e.        sin(x/2) = 1

 

            (x/2) =

 

            x =

 

At this point there is a point of inflection.  (must be a P.O.I. as graph is never decreasing)