Math 100 Midterm Exam: October 6, 2004, Exams of Calculus

The first midterm exam for math 100, taught by dr. Sandy rutherford, held on october 6, 2004. The exam covers topics such as rational and irrational numbers, solving equations, polynomial factorization, cubic equations, and function analysis. Students are required to show their work and provide reasons for their answers. No calculators, notes, or textbooks are allowed.

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Math 100 First Midterm
Instructor: Dr. Sandy Rutherford
Date: October 6, 2004
Time: 11:30 12:30
Name (please print):
Signature:
Student Number:
Read questions carefully, show all of your work and present organised solutions. If we do not
understand what you are doing, you will receive zero marks. You must give reasons for your
answers.
No calculators of any kind may be used.
No notes or textbooks may be used.
Some answers may contain radicals of the form 2. These should be simplified where possible;
however, a decimal expansion is not required.
This exam has 7 pages, including this cover page. Please be sure that you have all of the
pages.
Question Mark Total Marks
1 3
2 6
3 6
4 6
5 18
6 6
Total 45
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pf4
pf5

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Math 100 — First Midterm

Instructor: Dr. Sandy Rutherford Date: October 6, 2004 Time: 11:30 – 12:

Name (please print):

Signature:

Student Number:

  • Read questions carefully, show all of your work and present organised solutions. If we do not understand what you are doing, you will receive zero marks. You must give reasons for your answers.
  • No calculators of any kind may be used.
  • No notes or textbooks may be used.
  • Some answers may contain radicals of the form
  1. These should be simplified where possible; however, a decimal expansion is not required.
  • This exam has 7 pages, including this cover page. Please be sure that you have all of the pages.

Question Mark Total Marks

Total 45

  1. State the definitions of a rational and an irrational number.
  2. Solve for x in the following equations.

(a)

x + 3

  • x = 2

(b) x^2 =

x^2 − 2

  1. Consider the cubic equation

x^3 + x^2 + ax = 0 , where a is a real number.

Determine for which values of a this equation has 0,1,2, or 3 real solutions. Can it ever have more than 3 solutions? Explain.

  1. State the domain and range of each of the following functions. Also, find any roots (values of x at which the function is equal to zero) and state which functions are even or odd under reflection about the y-axis. Finally, sketch a reasonably accurate graph for each function.

(a) p(x) = x^4 + 2x^2 − 8

(b) q(x) =

x^2 + (^) x^1 x + (^) x^12

For this function, state the domain and range both before and after any continuous extension.

  1. A cell phone is situated on a straight line between two transmitting towers that are 10km apart. The received signal strength in watts from the first tower is 6 divided by the square of distance from the tower in km. The signal strength in watts from the second tower is 4 divided by the square of the distance from this tower in km. As the cell phone travels along the line from the first tower to the second, it will switch over to receiving signals from the second tower when the signal strength from the second tower equals and then exceeds the signal strength from the first tower. At what point on the line between the two towers will this occur?