DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCECOLLEGE OF SCIENCE

UNIVERSITY OF THE PHILIPPINES BAGUIO

Math 100: Introduction to Calculus - QUIZ 1

”Cheating is a form of intellectual dishonesty that will not be condoned. Anyone caught cheating shall be dealt with in

accordance with the rules on student conduct and discipline. Cheating is punishable with a grade of 5.0 in the course and/or

(1) suspension for a period of not less than one year; or (2) expulsion from the University.”

I. ALL or NOTHING. (2 pts each) State/Define the following precisely, give an example and explain.

1. a function f

2. domain and range of a function f

3. graph of a function f

4. vertical line test

5. domain of the composite function (f ◦ g)(x)

6. odd and even function

7. the two main goals of calculus (no example needed, just state and explain.)

II. ALL or NOTHING. (a, 3 pts) Sketch the graph of the following functions, and (b, 2 pts) find its domainand range.

a. f(x) = |x|·|5− x| b. f(x) = x + [|x|] c. f(x) =

x + 2 if x ≤ −4√

16− x2 if −4 < x < 42− x if x ≥ 4

III. ALL or NOTHING. (3 pts each) If f and g are two functions such that (f ◦ g)(x) = x and (g ◦ f)(x) = x,then f and g are inverses of each other. Show that f and g are inverses of each other.a. f(x) = 1/(1 + x) and g(x) = (1− x)/x b. f(x) = 2x− 3 and (x + 3)/2

IV. ALL or NOTHING. A page of print is to contain 24 sq. in. of printed region, a margin of 1.5 in. at thetop and bottom, and a margin of 1.0 in. at the sides. (a, 4 pts) Find a mathematical model expressing thetotal area of the page as a function of width of the printed portion. (b, 1 pt) What is the domain of yourfunction in part (a)?

V. ALL or NOTHING. The consumer demand for a particular toy in a certain marketplace is a function fof p, the number of pesos in its price, which in turn is a function g of t, the number of months since the toyreached the marketplace. If f(p) = 5000/p2 and g(t) = t2/20 + 7t/20 + 5, then do the following: (a, 4 pts)Find a mathematical model expressing the consumer demand as a function of the number of months sincethe toy reached the marketplace. (b, 1 pt) Find the consumer demand 5 months after the toy reached themarketplace.

VI. ALL or NOTHING. In a limited environment where A is the optimum number of bacteria supportableby the environment, the rate of bacterial growth is jointly proportional to the number of present and thedifference between A and the number present. Suppose 1 million bacteria is the optimum number supportableby a particular environment, and the rate of growth is 60 bacteria per minute when 100 bacteria are present.(a, 4 pts) Find a mathematical model expressing the rate of bacterial growth as a function of the number ofbacteria present. (b, 1 pt) What is the rate of growth when 100,000 bacteria are present?

VII. ALL or NOTHING. (2 pts each) Find the limit and when appropriate, indicate the limit theorems beingapplied.

a. limx→4

3x2 − 17x + 20

4x2 − 25x + 36b. lim

x→1

3√x− 1

x− 1c. lim

t→0

1−√

1 + t

t

VIII. ALL or NOTHING. If f(x) =

{x2 − 9 if x 6= 34 if x = 3

, find (a, 2 pts) limx→−3

f(x) and (b, 2 pts) show that

limx→−3

f(x) 6= f(−3).

- - -END- - - Perfect Score: 50 pts Highest Possible Score: 60 pts - - -END- - -

”If there is a better solution, find it!” -Thomas Edison

r.s.lagunero 29.Aug.2014

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