DAWSON COLLEGE
MATHEMATICS DEPARTMENT
Final Examination
Winter 2013
Calculus 1 (201‐103‐DW)
Date: Tuesday, May 14th, 2013 at 2:00pm
Instructors: C. Farnesi, A. Jimenez, N. Sabetghadam
1. [9 marks] Evaluate the following limits. Show all your work.
a.
2
15
lim
9
18
b.
lim
16
5
√
3
c.
lim
7
5
3
8
9
20
2. [4 marks] Given the piece‐wise function
find lim
defined as
3
3
15
3
, if it exists. Show all your work.
3. [6 marks] Given the piece‐wise function
1
determine if the function h
defined as
4
2
9
2
5
2
is continuous at
2. Show all your work.
4. [8 marks] Use the limit definition of the derivative (the four‐step process) to find the
derivative of the function
2
5
1
No marks will be given for using the rules of differentiation.
5. [5 marks] A company determines that the daily cost of producing a certain commodity
is given by the function
170
3081, and the selling price varies according to
the function
0.25
615, where is the number of items sold. Determine the
actual profit made on the 108th item.
6. [8 marks] Find the derivative of each of the following functions. Show all your work,
but do NOT simplify.
a.
√tan
b.
sin
7
3
7. [6 marks] Determine for which values of the function
3
4 2
5
has a horizontal tangent line.
8. [5 marks] Find an equation of the tangent line to the graph of the function
3
ln 2
when
1.
9. [8 marks] Consider the implicitly‐defined function
7
3
a. Find the derivative .
b. Find the slope of the tangent line at the point 2,1 .
10. [7 marks] Use logarithmic differentiation to find the derivative
cos
given that
11. [5 marks] Find the absolute extrema of the function
9
4
2
on the interval
5,0 .
12. [12 marks] Consider the function
4 .
a. Find the ‐ and ‐ intercepts of the function.
b. Determine the intervals where the function is increasing/decreasing and find the
relative extrema.
c. Determine the intervals where the function is concave up/ concave down and
find the inflection points.
d. Sketch the function.
13. [9 marks] An owner of a farm wants to enclose a rectangular area to grow apple trees.
Only 3 sides of this area will require fencing. If the farmer has exactly 86 feet of fencing
with which to enclose this area, find the dimensions of the maximum area possible for
growing apple trees.
14. [8 marks] Find the following anti‐derivatives:
a.
5 √
20
b.
7
2 7
1
3
Answers
1.
2.
3.
4.
5.
6.
B) 48
C)
12
is NOT continuous at 2 (because condition 3 fails).
4
5
The actual profit made on the 108th item is $391.50.
2
A)
tan
sec
√tan
A)
lim
No,
B)
7. The function has a horizontal tangent line when
8.
7
4
,
and
.
B) The slope of the tangent line is 8.
9. A)
2
10.
2
11. The absolute maximum is 2 and the absolute minimum is
.
12. A) The y‐intercept is 0,0 , and the x‐intercepts are 0,0 & 4,0 .
B) The function is increasing on the interval ∞, 0 0,3 and decreasing on 3, ∞. It
has a relative maximum at 3,27 .
C) The function is concave down on the interval ∞, 0 2, ∞ and is concave up on
0,2 . It has 2 inflection points, at 0,0 and 2,16 .
D)
y
30
20
10
-1
1
2
3
4
x
-10
13. The dimensions of the maximum area possible are 21.5 ft by 43 ft.
14. A)
5 ln| |
2√
20
B)
7
2
4