Preparing for University Calculus - Math and Computer Science

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3.6 Algebra with FractionsYou should be able to simplify a fractional expression, convert a “stacked” fractionalexpression into a simple one, put fractional expressions over a common denominator,and perform a partial fraction expansion. These skills will be useful in findingvarious derivatives, simplifying derivatives and integrals, and in particular for the“partial fractions” technique of integration.Examples:1. Simplify (x − 1)2 +4x.x +1Solution:(x − 1) 2 +4xx +1= x2 − 2x +1+4xx +1= x2 +2x +1x +1= x +1.2. Simplifyx−1x+2 − x−2x+1x.x−1x+2Solution:− x−2x+1x3x(x + 1)(x +2) .=(x−1)(x+1) − (x−2)(x+2)(x+1)(x+2)x= (x2 − 1) − (x 2 − 4)x(x + 1)(x +2)=3. Expand x2 +1x 2 in partial fractions.− 1Solution:x 2 +1x 2 − 1 =1+ 2x 2 − 1 =1+ 2(x + 1)(x − 1) .We want to write this as 1+ Ax+1 + B for suitable A, B. Putting the fractionsx−1over a common denominator, we have A(x−1)+B(x+1) = 0x+2; so A = −1,B = 1, andx 2 +1x 2 − 1 =1− 1x +1 + 1x − 1 ..20
Problems: (answers on page 40)11. Simplifyx +1 − 1x − 1 .2. Write x +1+1/x +1/x 2 in the form P (x)/Q(x) where P,Q are polynomials.3. Simplify x +1x +3 − x . Is this equal to 0 for any x?x +24. Simplify 1/x1/x 2 .1x5. Simplify− 1x+h.h6. Solve x +1x − 2 − xx − 1 =0.7. Simplify8. Simplify9. Expand1/x + 1/y1/x − 1/y .xx +1 − 1x(x +1) .1x 2 + x − 610. Expand x3 +4x 2 +4x +1x 2 +2x11. Expand1in partial fractions.in partial fractions.x 3 − x in partial fractions. 21
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: Problems: (answers on page 40)1. So
Page 23 and 24: Problems: (answers on page 40)1. Ra
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

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