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Simon Fraser University
MATH 154 โ MIDTERM 1
Instructor: D.Kent
October 5, 2005
Last Name____________________________
Given Name(s)________________________
Student ID____________________________
Signature____________________________
Question Maximum Mark
Total 40
INSTRUCTIONS
1. Do not open this booklet until
instructed to do so. The booklet contains
8 pages including the cover page.
- Print your name and student ID in the
space provided above.
- For each question you must show all
your work unless stated otherwise.
- No book, paper, or device other than the
usual writing instruments, this booklet, and
scientific calculators are allowed. In
particular, no graphing/programmable
calculators are allowed.
- During this examination, speaking to,
communicating with, or exposing written
papers to the view of other students is
forbidden.
- You may use the back of the previous
page for a rough work or if you run out of
space.
- Stop writing when you are instructed to
do so. Failure to follow instructions may
result in penalties.
- [5 marks total] Indicate whether the statement is True ( T ) or False ( F ). No
explanations necessary.
# Statement T F
1 The function^ f^ ( ) x^^ =^ x sin x^ is an odd function.
If g x ( ) is one-to-one function given by ( ) 3
x g x = + x + e , then
1 g (4) 0.
โ
Function f ( ) x is one-to-one if it successfully passes the horizontal
line test.
4 The derivative of a function f^ ( ) x^ is given by
f x h f x ) f x h
5 A rational function is continuous on its domain.
- [6 marks total] Find the following:
a) [1] The domain of f ( ) x =tan 2. x
- [4 marks] Draw a sketch of
2 3 10
x y
โ = โ
on a semilog plot and indicate the
values of intercepts.
- [4 marks total] The Monod growth function describes growth as a function
of nutrient concentration N :
r N ( )
N
r N N N
a) [2] Find the saturation level of this model.
b) [2] Find when the half-saturation is reached.
- [9 marks total] Find the following limits if they exist:
a) [2] 0
2
2
lim x
x
โ x
b) [2]
2 3 1 lim x 4
x x
โโโ x
- [4 marks] Use the definition of continuous function to show that the function
x f x (^) x
x
โช โ ^ โ
Is discontinuous at and identify the type of discontinuity. Sketch the graph
of the function.
x = โ 5
- [4 marks] State the Intermediate Value Theorem and use it to show that the
equation has a solution in the interval (-3, 1).
2 x โ 2 x โ 3 = 0
- [4 marks] Apply the definition of the derivative to find f '( ) x if
2 f ( ) x = 3 โ 2 x.
