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Chapter 5: Q. 3 (page 441)

What is a rational function? What does it mean for a rational function to be proper? Improper?

Short Answer

Expert verified

The function that can be expressed as a quotient of polyomial.

Step by step solution

01

Step 1. Given Information

The given term is Rational function

02

Step 2. Explanation

A rational function is a function that can be expressed as a quotient of polynomial.

A rational function is said to be proper if the degree of its numerator is strictly less than the degree of its denominator otherwise it is improper rational function.

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Most popular questions from this chapter

Explain why, if x=asinu, then a2−x2=acosu. Your explanation should include a discussion of domains and absolute values.

Solve the integral:∫xexdx

Why don’t we need to have a square root involved in order to apply trigonometric substitution with the tangent? In other words, why can we use the substitution x=atan⁡uwhen we see x2+a2, even though we can’t use the substitution x=asin⁡uunless the integrand involves the square root ofa2−x2? (Hint: Think about domains.)

Complete the square for each quadratic in Exercises 28–33. Then describe the trigonometric substitution that would be appropriate if you were solving an integral that involved that quadratic.

2(x+2)2

Solve ∫x+2x2+4x32dxthe following two ways:

(a) with the substitution u=x2+4x;

(b) by completing the square and then applying the trigonometric substitution x + 2 = 2 sec u.
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