Math 141Summer 2010

Midterm Exam Review

Topics (Percentages are Approximate)

• 2.1 Quadratics 10%• 2.2 Higher Degree Polynomials 5%• 2.3 Real Zeros of Polynomial Functions 10%• 2.4 Complex Numbers 5%• 2.5 The Fundamental Theorem of Algebra 10%• 2.6 Rational Functions and Asymptotes 10%• 2.7 Graphs of Rational Functions 10%• 3.1 Exponential Functions and Graphs 5%• 3.2 Logarithmic Functions and Graphs 5%• 3.3 Properties of Logarithms 10%• 3.4 Solving Exponential and Logarithmic Equations 10%• 3.5 Applications of Exponential and Logarithmic

Equations 10%

2.1 Quadratic Functions

• Standard Form of a Quadratic

• (h, k) is the vertex

• If a > 0 it opens upward.

• If a < 0 it opens downward

khxaxf 2)()(

Sample Problems 2.1• Sketch a graph of

• Write the equation in standard form. – Where is the vertex?– Where is the y-intercept?– Where are the x-intercepts?

• Write the standard form of the equation of the parabola that has a vertex at (6, -2) and passes through (8, -7)

3)6()( 2 xxf

14244)( 2 xxxf

Dog Walking

• The total revenue R (in dollars) earned by a dog walking service is given by

where p is the price charged per dog (in dollars)

a. Find the revenue when the price per dog is $4.

b. Find the price that will yield a maximum revenue.

pppR 15012)( 2

2.2 Polynomial Functions of Higher Degree

• Transformations of monomial functions

• Leading Coefficient Test

• Real Zeros (correspond to x-intercepts, factors (x – c), solutions to f(x) = 0)

• Repeated zeros (multiplicity)

2.2 Problems

• Sketch the graph of

• Describe the global (or end) behavior of

• Find a polynomial function with zero: -4 (multiplicity 1) and zero: 3 (multiplicity 3).

• Sketch a 5th degree polynomial with 2 real zeros and a positive leading coefficent.

5)3(2)( 3 xxf

xxxxf 1457)( 78

2.3 Real Zeros of Polynomial Functions

• Long division

• Synthetic Division

• Remainder Theorem

• Rational Zero Test

2.3 Problems• Divide

• Use the remainder theorem and synthetic division to find the value of

at x = 3

• Use the rational zero test to list all possible rational zeros of

Does this polynomial have any rational zeros?

)32()9614( 24 xxxx

xxxxf 1457)( 34

61457 34 xxx

2.4 Complex Numbers

• Standard Form

• Adding, subtracting, multiplying, dividing

• Complex conjugates

• Rationalizing the denominator

2.4 Problems

• Write in terms of i.

• Simplify each expression. Write your answers in a + bi form.

20

20

40

ii 3155 i52

2

ii 647 ii 21

2.5 The Fundamental Theorem of Algebra

• Linear factorization theorem

• Complex zeros

• Factoring a polynomial completely

2.5 Problems

• Find all the zeros of

• What are the x-intercepts of f(x)?

• Write the linear factorization of f(x)

xxxf 4)(

2.6/2.7 Rational Functions and Asymptotes

• Domain

• Horizontal and Vertical Asymptotes

• Holes

• Oblique (slant) asymptotes

• Sketching a graph

2.6 – 2.7 Rational Functions

• Find all holes, x-intercepts and asymptotes of

4

65)(

2

2

x

xxxg

• Identify all holes, x-intercepts and asymptotes of the function then graph it.

23

122)(

2

23

xx

xxxxf

Chapter 3: Exponents & Logarithms

• Graph

• How does this graph differ from the graph of ?

43)( xxg

xxf 3)(

• Evaluate without using a calculator.

• Find the exponential form of the logarithmic equation

9log3

4

38log16

• Evaluate using the change of base formula.

• Use the properties of logarithms to condense the logarithmic expression.

5

1log8

)1(log235log 44 x

• Solve for x.

• Solve for x.

• Solve for x.

• Solve for x.

01lnln x

2)144(log8 x

892 xx

523 255 xx

Find the time required for an investment of $25000 to double if the annual interest rate of 10% is compounded monthly.

The half-life of carbon-14 is 5700 years. Suppose you start with 100 grams of carbon-14.

a) Find a model for the amount left after t years.

b) How much is left after 1000 years?

c) How long until only 33 grams remain?

• A rumor begins at a closed-campus high school that the principal has donated one million dollars for the graduation party. There are 2100 students in the high school and the spread of the rumor is modeled by

where y is the number of students who have heard the rumor and t is the time in minutes.

a) Where are the horizontal asymptotes and what do they represent?

b) How long until 40% of the students have heard the rumor?

tey

7.020991

2100