Ch.2: Polynomials and Rational Functions

Precalculus H/GT

Spring Semester Final Exam Review

Ch.2: Polynomials and Rational Functions:

ü Long and Synthetic Divisions to find roots, quotients, remainders, factors, slant asymptotes, etc.

ü Remainder Theorem

ü Writing Polynomials with given roots

ü Finding zeros, horizontal and vertical asymptotes of rational functions

ü Solving polynomial and rational inequalities using sign charts

Ch.3: Exponential and Logarithmic Functions:

ü Exponential and logarithmic graphs (including domain, range, asymptotes, intercepts, etc.)

ü Inverse function relationship between exponential and logarithmic functions

ü Properties of Logarithm expressions (expanding and condensing logarithm expressions)

ü Base Change Formula

ü Solving exponential and logarithm equations

ü [Calc] Compound Interest (simple, compound monthly, quarterly, annually, continuously, etc.)

ü [Calc] Growth and Decay (exponential growth and decay, half life, etc.)

Ch.10: Conics

ü Knowing standard forms of four conic shapes

ü Converting from general form to standard form

ü Characters of each conic shape (center, foci, axes, vertices, eccentricity, etc.)

Ch.10: Polar

ü Conversion between polar and rectangular coordinates and points

ü Conversion between polar and rectangular equations

ü Identifying and graphing conic shapes (four types of limacon, spiral, lemniscates, roses, etc.)

Ch.10: Parametric

ü Conversion between rectangular and parametric equations

ü Parametric equations for four conic shapes

ü [Calc] Applications of parametric equations (projectile, ferries wheel, etc.)

Ch.9: Sequences and Series

ü General sequences and series with pattern recognizing

ü Sigma Notation for series

ü Arithmetic Sequences and Series (finding common difference, number of terms, sum, etc.)

ü Geometric Sequence and Series (finding common ration, number of terms, sum, etc.)

ü Infinite Geometric Series (finding the sum and identifying convergence and divergence)

ü [Calc] Applications of Sequences and Series

Ch.9: Counting Principles, Probabilities and Binomial Theorem

ü Counting Principles (arrangements, factorials, permutations, combinations, etc)

ü Probabilities

ü Binomial Theorem (finding coefficients of terms from a binomial expansion)

Practice Problems

Ch.2: Polynomials and Rational Functions

1. Write a polynomial with lowest degree, if and –4 are two of the roots of the polynomial.

2. Given that has –2 as one root. Use synthetic division to find other roots.
roots: –2 (multiplicity of 2), 5/2

3. Given that is a factor of the polynomial , find the value of k.
–17

4. Find the slant asymptote of the rational function .

5. Solve the inequality .

6. Find the interval(s) where the rational function is non-negative.

Ch.3: Exponential and Logarithmic Functions

7. Consider the two functions and . Find the domain, range, x and y-intercepts and asymptotes of these two functions.
: domain all real numbers, range all positive real numbers, y-int at , horiz asy: x-axis
: domain positive real numbers, range all real numbers, x-int at , vert. asy: y-axis

8. Write the logarithm expression in the corresponding exponential form.
, and

9. Write the exponential expression in the corresponding logarithm form.

10. Find the inverse function of .

11. Expand the logarithm expression .

12. Condense the logarithm expression .

13. Use Base Change Formula to express using: (a) common log; (b) natural log; (c) log base 3.

14. Solve the equation .

15. Solve the equation .

16. [Calc] A bank offers an annual interest rate of 5% compounded quarterly, then how much money is needed to deposit in order to receive $25000 in 8 years? What if the interest rate was 5% compounded continuously?

17. [Calc] Half life of a radioactive element is 5 years. If 55 grams of this element is found today, then how much will it remain after 12 years? How many years ago was the mass 100 grams?
10.421 grams; 4.312 years

18. [Calc] The population of a certain bacteria grows exponentially. If population takes 10 days to grow from 200 to 900, then how many days will it take to grow from 900 to 3000?
8.005 days

Ch.10: Conics

19. Determine the shape of the graph produced by the equation . Graph it and find its focus and eccentricity
, parabola, Focus , eccentricity = 1

20. Graph , and find its foci and eccentricity.
, Focus , eccentricity =

21. Graph , and find its foci and eccentricity.
, Focus approximately , eccentricity

22. Find the equation of a circle if the end points of a diameter are and .

Ch.10: Polar

23. Convert the polar points to rectangular coordinates: .

24. Convert the following rectangular points to polar points: , [Calc].

25. Convert the following polar equations to rectangular equations: , .

26. Convert the following rectangular equations to polar equations: , .

27. Graph the following polar equations, and name the graphs:
Circle, dimpled, inner loop, cardioid, convex, rose, lemniscates, spiral of archimedes
[graph: check with calculator]

Ch.10: Parametric

28. Convert the following parametric equations to rectangular equations and graph them:

29. Write a parametric equation for a circle centered at with radius 4. Make the starting point, going counterclockwise.

30. Write a parametric equation for the rectangular equation .

31. Write a parametric equation for the rectangular equation .

32. Write a parametric equation for the rectangular equation .

33. [Calc] Write a parametric equation for an arrow being released from the origin at an angle of elevation of with an initial speed of 36 ft/sec. Use the parametric equation to find the position of the arrow after 3 seconds. Assume the acceleration due to gravity is –32 ft/sec2.

34. [Calc] A Ferris wheel has a diameter of 90 meters long and its highest point is 93 meters above the ground. The Ferris wheel rotates clockwise at a speed of 13 minutes per revolution. If Mary starts her ride on the Ferris wheel at the lowest point, find a parametric equation that represents Mary’s position at any time t in minutes. What will be her height above the ground after 3.5 minutes?

Ch.9: Sequences and Series

35. The first term and the 7th term of an arithmetic sequence are 18 and 36, respectively. Find the 19th term of the sequence.
69

36. An auditorium has 20 rows of seats. The front row has 45 seats. Each row after that has 4 more seats than the previous row. Find the total number of seats in the auditorium.
1660

37. Evaluate: , , ,
1118, diverges, 68,

38. [Calc] In the January last year, 300 zoo tickets were sold. The zoo ticket sale increased by 20% each month after the first month. To the nearest integer, find the number of zoo tickets sold in last year.
7788

39. The first term of a geometric sequence is while the 6th term is . Find the common ratio.
3/2

Ch.9: Counting Principles, Probability and Binomial Theorem

40. Evaluate: , ,
210, 56, 15

41. [Calc] How many ways can the letters in “MISSISSIPPI” be arranged?
34650

42. In how many ways can 7 flags of different colours be placed around a circular track field?
720

43. In how many ways can a necklace be made using 7 pearls of different colours?
360

44. [Calc] How many ways can 10 letters, A to J, be arranged in a row, if A and B must be adjacent?
725760

45. [Calc] Ten letters, A to J, are randomly arranged in a row. What is the probability that A and B are adjacent?
0.2

46. [Calc] There are 15 red marbles and 12 blue marbles in a box. If 5 marbles are randomly chosen, what is the probability that exact three are red?
0.372

47. [Calc] Five cards are dealt from a deck of 52 cards. What is the probability that the five cards all have the same suit?
0.00198

48. What is the coefficient of the term in the expansion of ?
–1080