DIVISION OF LIBERAL ARTS AND HUMAN SERVICESDIVISION OF LIBERAL ARTS AND HUMAN SERVICESMATHEMATICS DEPARTMENT

FINAL EXAMINATION

MATH 118: Pre Calculus

DATE: Thursday , 26th July, 2012

TIME: 3 hours - 5:00-8:00 pm

PERIOD: Trimester II - 2011/2012 Academic Year

LECTURER K. DANIELCRN# 36083

INSTRUCTIONS:

This paper consists of SIX questions

You are required to:

Show all necessary working

Answer ANY FOUR questions.

The use of a non-programmable electronic calculator is allowed

Do not write on this paper

This paper has 5 pages

Additional sheets of ruled paper will be provided upon request

DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO

College of Science, College of Science, Technology and Applied ArtsTechnology and Applied Arts

of Trinidad and Tobagoof Trinidad and Tobago

1. a. Given the function g( x )=x2 ( x−4 )(2 x+6) :

i. Determine the end behavior: find the power function that the graph of resembles for large values of |x|.

[2 marks]

ii. Find the x- and y-intercepts of the graph. [4 marks]

iii. Determine whether the graph crosses or touches the x-axis at each x-intercept.

[3 marks]

iv. Sketch the graph of g(x), labeling the intercepts. [5 marks]

b.

Write a possible polynomial function to represent the graph given below.

[4 marks]

c. The function m( x )=x3+ px 2+qx+6 has the same

remainder when divided by ( x+1)and(2−x ). Given that the remainder when the expression is divided by ( x+3) is -60, find the values of p and of q.

[7 marks]

2. a. Solve the following equations:i.

4 x−3=16x−1 [4 marks]

ii. log 2( x+4 )+log2 ( x−2)=4 [6 marks]

b.

The logistic growth function f ( t )=25 ,000

1+356.1 e−1.8t models the number

of people who have become ill with swine flu t weeks after its initial outbreak in Tobago.

i. How many people became ill with this infection when the epidemic began?

[3 marks]

ii. How many people were ill after 3 weeks? [2 marks]

iii. What is the maximum number of people that can be expected to fall ill?

[1 marks]

c. Sandy manages a ceramics shop and uses a 650ºF kiln to fire ceramic green-ware. After turning off her kiln, she must wait until its temperature gauge reaches195ºF before opening it and removing the ceramic pieces. If room temperature is 80ºF and the gauge reads 500ºF in 11 minutes, how long must she wait before opening the kiln? Round your answer to the nearest whole minute.

Assume the kiln cools according to Newton's Law of Cooling where, T a

is the temperature of the room,T 0 is the initial temperature of the kiln, and t is the time, in minutes.

T ( t )=T a+(T0−T a )ekt

[9 marks]

3. a.

Find the partial fraction decomposition of the following rational expressions:

[10 marks]

b i. Solve the system of equations using elimination.3 x2+2 y2=85x2−2 y2=−21

[8 marks]

ii. The area of a rectangular garden is 6000 square meters, and the length of its diagonal is 130 meters. Find the dimensions of the garden.

[7 marks]

4. a.

Find the domain of the rational function:

[3 marks]

b.

A closed box with a square base, with length r and height h, has to have a volume of 10,000 cubic meters. i. Find a function for the surface area of the box in terms of V(r). [3 marks]

ii. What is the horizontal asymptote for the function V(r)? [2 marks]

iii Describe the horizontal asymptote of V(r) in practical terms. [1 marks]

22

3

6

3142

x

xx

954)2(2)( 2

xxxxxf

.c.

Sketch the graph of the rational function Make sure to include: Any asymptotes x- and y-intercepts Any behavior that the graph may exhibit.

[16marks]

5. a.

A bridge is built in the shape of a parabolic arch. The bridge arch has a span of 180 meters and a maximum height of 30 meters. Find the height of the arch at 20 meters from its center.

[8 marks]

b.

c

Write the general equation of the conic represented in the graph below.

Identify the center, foci, and vertices of the ellipse described by the equation

[7 marks]

[10 marks]

6. a) A sequence of real numbers satisfies .

Find the common ratio and write the first four terms of this sequence [3 marks]

b) Using the binomial theorem, expand the given binomial and simplify:

c) Find the sum of the series ∑r=1

20

( r2−2r+3 )

d) A theater has 20 rows with 24 seats in the first row, 28 in the second row, 32 in the third row, and so forth. How many seats are in the theater?

e) A bicycle wheel rotates 300 times in a minute as long as the rider is pedaling. If the rider stops pedaling, the wheel starts to slow down. Each minute it will rotate only 2/ 3 as many times as in the preceding minute. How many times will the wheel rotate in the 4th minute after the rider's feet leave

8 marks]

[ 6 marks]

[ 4 marks] [ 4marks]

2)3(1x

nu 11 3,135 nn uuu

the pedals? Round your answer to the nearest unit.

END OF EXAMINATION