Expert: Scotto - 11/7/2008

Question
1.  f(x)=sqrt(x-1)
   
   a)  Draw a graphical representation of the method or process of finding the derivative of this function at any point x.  Show the definition of the derivative graphically.

   b)  Please explain the method used to find the derivative of a function in this example.  Make sure your explanation is related to the definition of the derivative.

   c)  Find the derivative of this fuction at any point x using the definition of the derivative.

   d)  Let (3+tan(x))/(sin(x)).  Write the equation of the normal line to f(x) at x=pi/4.  Please give exact answers and no decimals.


Answer
(a) I don't yet have a scanner to scan this in, so I can't do (a).  I can tell you it is similar to a sideways parabola with the leftmost point at (1,0).  It curves out equally above and below this point to the right.

(b) What we really have is f(x) = (x-1)^0.5.  This is a function of x to a power, and these are all done by multiplying the function by the exponent, reducing the exponet by 1, and multiplying by the derivative of the inside.

The exponet in this case is 0.5 and the derivative of the function inside is dx.

(c) The result is f'(x) = [0.5(x-1)^(0.5-1)]dx = [(x-1)^-0.5 / 2] dx

(d) To find the tangent line, we need to find the derivative of the function since that tells us what the slope is at that point.
That function that is being differentiated is f(x)/g(x) where
f(x) = 3 + tan(x) and g(x) = sin(x).

To differentiate a quoetient of two functions, the answer is
(lo d hi - hi d lo)/lo², or, (g*f' - f*g')/g².
f'(x) = sec²(x) and g'(x) = cos(x).  The functions f(x) and g(x) were given in the last paragraph.  Put the functions and their derivates into the equation given and simplify as necessary.
It would be good to note that tan(x) = sin(x) / cos(x) and that
sec(x) = 1 / cos(x).