Math 103 Final Fall 2001

Show all work. No credit will be given without proper mathematical justification.

All questions are worth 10 points as indicated within each problem.

1. a) The polynomial can be expressed in the

form (x+2) (x). Find (x).

b) Find the roots of the expression and the use the Root Factoring

Method to express as the product of its factors. If possible,

remove any fractions from the final factorization.

2. a) Find a system of 3 equations in 3 unknowns that can be used to find the

equation of the parabola with vertex (1/2 , 2), with an Axis of Symmetry parallel

to the X-axis and such that the point (1, 3) lies on the parabola (1, 3). DO

NOT SOLVE the system of equations!!

b) Solve the following system of equations for a, b and c:

3. a) If =19, find the value of x.

b) If = 10, there are 2 consecutive integers, m and n, such that m < x < n.

Find the values of m and n respectively.

c) Solve for x: ln (2 - x) = 2.

d) For the function y = ln find the domain, the vertical asymptotes,

and the behavior of the function as it approaches the boundaries of the domain.

4. Verify the change of base formula for as follows:

a) Write an equation that equates the number 8 to an exponential expression in

which the base is 2 and the exponent is a logarithm. Put the 8 on the left hand

side of the equation.

b) In the equation obtained in part a), replace the number 8 on the left hand side

with an exponential expression in base . On the right hand side, replace the

base 2 by an exponential expression in base .

c) Simplify the right hand side of the equation using the algebraic rule that states

"when raising a power to a power, multiply the exponents ".

d) Since the expressions on each side of the equation have the same base take the

natural logarithm of each side of the equation and use the Log Rule for Logs:

ln .

e) Put this equation into the form that verifies the Change of Base Rulefor Logs.

5. a) Find only the x-coordinate of the critical point of the function

y =

b) Find only the x-coordinate of the point of inflection of the curve whose domain

is 2 < x < and whose first derivative is y' = and then indicate the

intervals on the x-axis for which the the curve is concave up or concave down.

Show a mathematical justification for your answer.

6. a) Find the domain of the function y =] and indicate the

domain in a diagram on the x-axis. 3

b) Solve for the values of x that satisfy the inequality and

represent them graphically on the x-axis. (The problem may be done

analytically or geometrically). 3

c) Find the domain of the function y =

7. a) Solve for y in terms of x:

b) Solve for all values of y:

8. a) Sketch the graph of x using the calculus method, and then

find the functions of the form y = f(x) that represent the "top" and

"bottom" halves of the parabola.

b) Solve for x and check your answers in the original equation to determine if

there are any solutions to this equation, and, if so, what they are

9. a) A rolling wheel of radius 10 inches is making 40 revolutions per minute

(RPM). Find the speed at which the center of the wheel is rolling in miles

per hour (MPH). (The answer should be left as an arithmetic fraction.)

b) A wheel of radius 13 inches rolls 2 miles. How many degrees does a spoke on

the wheel turn through and how many revolutions does the wheel make?

c) Find the Cartesian coordinates of point P if the distance from the origin, O, is 2

and the angle of inclination of OP is . (Show a special right triangle in a

quadrant diagram to justify the evaluation of trigonometric functions. )

10. Using the Laws of Logarithms, verify that if , then

satisfies this equation.