"Do the examples above bring back old (fond?) memories of LIMITS?"
Mult. by 1
Mult. by 1
Before we comment on the algebra, let's examine these
examples in a cal-
culus context. Since our virtual calculus site is still in development, you'll
need paper and pencil.
Exercise 1: Evaluate the following.
Note that algebra is essential in evaluating the above
limits and integrals.
Another way to evaluate would be as follows:
Just for fun (we're not afraid to use fun and calculus in
the same sentence),
show both answers to Exercise 1(g) are equivalent.
A. Long Division
A fraction ( rational expression ) that has a monomial denominator , like
Example 4, easily separates. However, if the fraction is improper (degree of
the numerator is greater than or equal to degree of the denominator), then
dividing by a polynomial requires Long Division. Here's Example 5 worked
B. Multiplying by "1"
Of our original examples, three of them (#2, 6, and 7) fall into this category.
The integral for sec x is derived using multiplication by a form of 1.
C. Completing the Square
In your previousmath courses , completing the square wasused to solve
quadratics or to change the form of a second degree polynomial of the type
Ax2 + By2 + Cx + Dy + E = 0. In Calc II, completing the square will be a
useful (and necessary) tool in evaluating certain integrals.
Un like our first example, substitution won't work. Let's
complete the square
and see what follows.
Conclusion: This review topic (as well as Topics 3 and 4)
how algebra is used to evaluate limits and integrals. The challenge is not
just knowing the algebra, but being able to recognize when to apply it. That
only comes from practice.
2.1 Find the quotient and remainder:
2.2 All of the following problems require multiplication by a form of 1
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