Karl's Calculus Tutor - Remedial Exercises

1) Find the sum of he following polynomials:

   x³  +  3x²  -  4x  +  6

and

   x⁴  -  7x³  -  6x²  +  x  +  7

Remember that you have to line terms of like powers and add them.

2) Find the product of the following:

   (x - 7) ×  (x²  -  5x  +  4)

Remember that you apply the distributive law to multiply this -- that is, multiply x by all the terms in the right-hand parentheses, then multiply -7 by all the terms in the right-hand parentheses, then gather up terms of like powers and add them.

3) Use the distributive law to expand into polynomials:

   (x + 1)²

   (x + 1)³

   (x + 1)⁴

Now do the same for:

   (x - 1)²

   (x - 1)³

   (x - 1)⁴

There is a pattern in the coefficients that we will discuss at length in a later section.

4) Suppose that:

   f(x) = x²  -  5x  +  4

and

   g(x) = x  -  1

Write expanded polynomial expressions for the following:

   f(x + 1)

   f(x - y)

   f(x) + g(x)

   f(x) × g(x)

   f(g(x))

   g(f(x))

   f(g(x + 2) + 1)

5) If you have a function, y = f(x), then the function that takes a value of y backward to the x value that f(x) used to produce it is called the inverse function of f(x). So, for example, square root is the inverse function of y = x². So we can say, in this case, that sqrt(y) = x because the functions are inverses of each other.

Not every function has an inverse. And even those that have inverses often have inverses that are not defined everywhere. For example, the inverse of y = x² is not defined for negative x. A function, y = f(x) has an inverse at some value, x only if the y value produced by f(x) cannot be produced by applying f to any other x. With the example of the x² function, we can only guarantee that if we exclude negative values of x. That is because x² = (-x)².

The standard notation for the inverse function of y = f(x) is:

   f^-1(y) = x

Some functions are their own inverses. An example of this is:

              1
   y = f(x) =  
              x

If you define that to be f(x), then write an expanded expression for:

   f^-1( f(a) + f(b) )

If you end up with something that has fractions in the denominator, see if you can massage it into something equivalent that does not have fractions in the denominator. For the record, the way of combining a and b that you have just written (assuming you did it right) is called the harmonic sum.

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