Final Exam Review

MATH 111 Final Exam Review Exercises

Overview

  • The following exercises are designed to help students prepare for the Math 111 final exam.

  • It is recommended that students also review past recitation quizzes and relevant textbook chapters (Chapters 1, 2, and 3).

  • The exercises provide practice but are not an exhaustive representation of potential exam problems.

  • Only exact answers are accepted during the exam, and all calculations should be shown to receive credit.

Exercise Categories

1. Factoring

  • Problem 1: Factor completely.
      - (a) (x−1)(x+2)2−(x−1)2(x+2)(x - 1)(x + 2)^2 - (x - 1)^2(x + 2)
      - (b) (a2+1)2−7(a2+1)+10(a^2 + 1)^2 - 7(a^2 + 1) + 10
      - (c) 3x−1/2+4x1/2+x3/23x^{-1/2} + 4x^{1/2} + x^{3/2}
      - (d) (x−1)7/2−(x−1)3/2(x - 1)^{7/2} - (x - 1)^{3/2}
      - (e) x−1/2(x+1)1/2+x1/2(x+1)−1/2x^{-1/2}(x + 1)^{1/2} + x^{1/2}(x + 1)^{-1/2}

2. Operations and Simplification

  • Problem 2: Perform operations and express answers in simplest form with positive exponents only.
      - (a) x^{-5} igg( rac{4x^{-2}y^3}{3x^{-5}} igg)^{-2}
      - (b) 2y^3 igg( rac{12x^3y^{-2}}{3x^{-5}y^4} igg)^{1/3}
      - (c) 3(1+x)1/2−x(1+x)−1/2(1+x)3(1 + x)^{1/2} - x(1 + x)^{-1/2}(1 + x)

3. Simplifying Expressions

  • Problem 3: Perform operations and express the answer in simplest form.
      - (a) racx−5x2−2x−8−racx+1x+2rac{x - 5}{x^2 - 2x - 8} - rac{x + 1}{x + 2}
      - (b) racx−43x3−27−rac2x+1x−3rac{x - 4}{3x^3 - 27} - rac{2x + 1}{x - 3}

4. Fraction Reduction

  • Problem 4: Express the following as a simple fraction reduced to lowest terms:
      - rac1−4z+4z2z2−2z3rac{1 - 4z + 4z^2}{z^2 - 2z^3}

5. Solving for Variables

  • Problem 5: Solve for yy in the equation:
      - rac1y+rac1x=rac1zrac{1}{y} + rac{1}{x} = rac{1}{z}

6. Radical Simplification

  • Problem 6: Simplify the following radicals (leave in radical form).
      - (a) 3<br>oot354x5y9z43<br>oot{3}{54x^5y^9z^4}
      - (b) rac<br>oot24x3y2z5<br>oot12xyz2rac{<br>oot{24}{x^3y^2z^5}}{<br>oot{12}{xyz^2}}
      - (c) <br>oot3x2<br>ootx<br>oot{3}{x^2}<br>oot{x}

7. Rationalizing Denominators

  • Problem 7: Rationalize the denominator and simplify.
      - (a) rac8x<br>oot32xy2rac{8x}{<br>oot{3}{2xy^2}}
      - (b) rac6<br>oot10−<br>oot7rac{6}{<br>oot{10} - <br>oot{7}}
      - (c) racx−3<br>ootx−<br>oot3rac{x - 3}{<br>oot{x} - <br>oot{3}}
      - (d) rac3h<br>ootx−h−<br>ootxrac{3h}{<br>oot{x} - h - <br>oot{x}}

8. Solving Equations

  • Problem 8: Solve for exact real value(s) of xx:
      - (a) rac4x−1+rac2x+1=35rac{4}{x - 1} + rac{2}{x + 1} = 35
      - (b) x4/3+8=0x^{4/3} + 8 = 0
      - (c) 6y2−12y=−36y^2 - 12y = -3
      - (d) x4−2x2−24=0x^4 - 2x^2 - 24 = 0
      - (e) rac23∣2x−3∣+4=10rac{2}{3}|2x - 3| + 4 = 10

9. Solving Inequalities

  • Problem 9: Solve the inequalities; represent solutions in interval notation.
      - (a) 3|3x - 2| + 1 < 22   - (b) |3x - 4| - 1 ext{ } 2   - (c) 2x^2 - 13x > -15
      - (d) 3x - 1 ext{ } 2x + 2

10. Value of Entertainment System

  • Problem 10: An entertainment system was purchased for $3,000 in 1997. In 1999, it was worth $1,500.
      - (a) Express the value vv as a function of time tt in years.
      - (b) Predict the value in 2000 using your function.
      - (c) Determine the year the system will be worth nothing.

11. Midpoints and Circles

  • Problem 11: Given points A(3, 1), B(-1, 5), C(4, 5), and D(2, 7):
      - (a) Find the distance between midpoints M (AB) and N (CD).
      - (b) Equation of the circle with center M and radius 5.
      - (c) Equation of a circle with center A, containing point B.
      - (d) Equation of a circle with a diameter having endpoints C and D.

12. Investment Problem

  • Problem 12: Jason invested in two investments: a CD earning 4% per year and a bond earning 8%. He invested $3,520 more in the CD than the bond, earning a total of $1,100.80 in annual interest. Calculate investments in both.

13. Function Evaluations

  • Problem 13: Given graphs of f(x) and g(x):
      - (a) Evaluate (f+g)(2)(f + g)(2).
      - (b) Evaluate f(g(1))f(g(1)).
      - (c) Solve f(x)=g(x)f(x) = g(x).
      - (d) Determine intervals where f(x) > g(x).
      - (e) Find f−1(6)f^{-1}(6).

14. Function from Table and Graph

  • Problem 14: Given f(x) defined by a table and g(x) by a graph:
      - (a) Evaluate f(2)f(2).
      - (b) Evaluate (f ullet g)(2).
      - (c) g(f(1))g(f(1)).

15. Compositions of Functions

  • Problem 15: Given f(x)=4xf(x) = 4x and g(x)=<br>ootx+4g(x) = <br>oot{x} + 4:
      - (a) Domain of (f+g)(x)(f + g)(x),
      - (b) (f ullet g)(x) and its domain,
      - (c) (g ullet f)(x) and its domain.

16. Function Evaluations and Limits

  • Problem 16: Given f(x)=2x2−x+1f(x) = 2x^2 - x + 1:
      - (a) Evaluate f(−2)f(-2).
      - (b) Find and simplify racf(x+h)−f(x)hrac{f(x + h) - f(x)}{h}.

17. Function Evaluations with Rational Expressions

  • Problem 17: Given f(x)=rac5xx+1f(x) = rac{5x}{x + 1}:
      - Find and simplify racf(x+h)−f(x)hrac{f(x + h) - f(x)}{h}.

18. Domain Determination

  • Problem 18: Find the domain of the following functions (show analysis):
      - (a) f(x)=<br>oot4−x2f(x) = <br>oot{4 - x^2}.
      - (b) f(x)=rac<br>ootx+3x−4f(x) = rac{<br>oot{x + 3}}{x - 4}.

Solutions Overview (Page 10)

1. Factoring Solutions

  • (a) 3(x−1)(x+2)3(x - 1)(x + 2)

  • (b) (a−2)(a+2)(a−1)(a+1)(a-2)(a+2)(a-1)(a+1)

  • (c) (x+3)(x+1)x(x−1)3/2(x + 3)(x + 1) x(x - 1)^{3/2}

  • (d) (x−2)(x - 2)

  • (e) (2x+1)(2x + 1)

2. Simplification Solutions

  • (a) rac253x8/3rac{25}{3}x^{8/3}

  • (b) y3+2x(1+x)3/2y^3 + 2x(1 + x)^{3/2}

  • (c) −x2+4x−1-x^2 + 4x - 1

3. Expressing as Simple Fractions

  • (a) rac−6x3−21x2−8x−43x(x−3)(x+3)rac{-6x^3 - 21x^2 - 8x - 4}{3x(x - 3)(x + 3)}

4. Fraction Reduction

  • z(z−2)z(z - 2)

5. Solving for y

  • y=racxzx−zy = rac{xz}{x - z}

6. Radical Simplification

  • (a) 3xy3z<br>oot333xy^3z<br>oot{3}{3}

  • (b) rac12x2yz3<br>oot62yzrac{12x^2yz^3<br>oot{6}}{2yz}

  • (c) xx

7. Rationalizing Denominators

  • (a) 4<br>oot34x2y4 <br>oot{3}{4x^2y}

  • (b) rac2<br>oot10+2<br>oot7yrac{2<br>oot{10} + 2<br>oot{7}}{y}

  • (c) 3−3<br>ootx−h−<br>ootx3 - 3<br>oot{x} - h - <br>oot{x}

8. Solving Equations Result

  • (a) x=rac112x = rac{11}{2}
    (b) More numbers solve the equations

All problems should be worked out in detail to show full understanding and receive complete credit.

Conclusion

  • The above review exercises and solutions should be studied extensively in preparation for the Math 111 final exam. Attention should be focused on calculations, graphing, factorization techniques, and function analysis as they comprise critical components of the curriculum leading up to the exam.