Preparing for University Calculus - Math and Computer Science

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3.7 Rationalizing numerators or denominatorsYou should know how to eliminate square (and other) roots from the numerator ordenominator of a fraction by multiplying both the numerator and denominator byan appropriate expression. This technique will be important in finding the derivativesof certain expressions involving roots.Examples:11. Rationalize the denominator of √ aSolution: By multiplying both numerator and denominator by √ a weobtain1√ a= 1 √ a√ a√ a=2. Rationalize the denominator of 1+√ x2+ √ xSolution: Recall, when we multiply (a + b)(a − b) we obtain a differenceof squares a 2 − b 2 . So, if the denominator of a rational expression containsa constant added to a square root term, we can eliminate the root term bymultiplying by the difference of the constant and the root term. In this caseif we were to multiply both the numerator and the denominator by 2 − √ xwe will obtain the desired result.1+ √ x2+ √ x = (1 + √ x)(2 + √ (2 − √ x)x) (2 − √ x)= 2 − √ x +2 √ x − x4 − 2 √ x +2 √ x − x= 2+√ x − x4 − x3. Rationalize the numerator of 1+√ x2+ √ xSolution: This time we multiply numerator and denominator by theconjugate of the numerator:1+ √ x2+ √ x = (1 + √ x)(2 + √ (1 − √ x)x) (1 − √ x) = 1 − x2 − √ x − x22√ aa
Problems: (answers on page 40)1. Rationalize the denominator of x √ +1x2. Rationalize the denominator of3. Rationalize the denominator ofa√ a3 √ b1√ x +√ y4. Rationalize the denominator of a − √ xb − √ x15. Rationalize the denominator of1+x 1/3 + x 2/3√ √ x + y6. Rationalize the numerator ofx7. Rationalize the numerator of a − √ xb − √ x8. Rationalize the numerator of9. Rationalize the numerator of√ √ x +1− xx3√ √ y + h −3yh23
Page 1 and 2: Preparing for University CalculusPr
Page 3 and 4: 1 About First-Year Calculus1.1 What
Page 5 and 6: 1.5 What is taking university calcu
Page 7 and 8: • Your university or student unio
Page 9 and 10: 2 Ordering informationGetting a cop
Page 11 and 12: Problems: (answers on page 40)1. Fi
Page 13 and 14: Problems: (answers on page 40)1. Si
Page 15 and 16: Problems: (answers on page 40)1. If
Page 17 and 18: Problems: (answers on page 40)1. If
Page 19 and 20: Problems: (answers on page 40)1. So
Page 21: Problems: (answers on page 40)11. S
Page 25 and 26: Problems: (answers on page 40)1. Fi
Page 28 and 29: 3.10 Exponents and rootsYou should
Page 30 and 31: 3.11 LogarithmsLogarithms are extre
Page 32 and 33: 3.12 Geometry and basic trigonometr
Page 34 and 35: 3.13 Trigonometric identitiesThere
Page 36 and 37: 3.14 Problem solvingSolving applied
Page 38 and 39: 3.14159265358979... Harder Problems
Page 40 and 41: Answers to problems:Section 3.1 (1)
Page 42 and 43: Some useful facts:Algebraic identit

Preparing for University Calculus - Math and Computer Science

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