MATH 1210-6
Spring 2003
Midterm exam II
Student Name:
Student ID Number:
Course Abbreviation and Number:
Course Title:
Instructor:
Math 1210
Calculus I
Vladimir Vinogradov
Date of Exam:
Time Period:
Duration of Exam:
Number of Exam Pages:
(including this cover sheet)
Exam Type:
Additional Materials Allowed:
March 7, 2003
Start time: 12:55 pm
1 hours
8
Closed Book
Calculator
Marking Scheme:
QUESTION VALUE SCORE
1
70
2
30
TOTAL
100
*) The bonus question counts for 10 points maximum.
End Time: 1:55 pm
1. (70 points) Derive enough information about the curve and sketch the graph of the following
function. Show a) all asymptotes, b) all critical values, c) all points of inflection.
y(x) =
x2 − 4
.
x2 − 1
2
3
10
8
6
4
2
-3
-2
-1
1
-2
4
2
3
2. (30 points) Find the absolute maximum and minimum of the function
y = x3 − 12x2
on the interval −2 ≤ x ≤ 15
5
6
√
Bonus question (10 points). A rectangle is to be inscribed in a semicircle of radius r (y = r2 − x2 ),
as shown in the figure. What are the dimensions of the rectangle if the area is to be maximized?
ANSWER:
7
Useful formulae
Constant multiple rule:
(k f (x))0 = k f 0 (x)
Sum rule:
(f (x) + g(x))0 = f 0 (x) + g 0 (x)
Product rule:
(f (x)g(x))0 = f 0 (x)g(x) + f (x)g 0 (x)
Quotient rule:
Chain rule:
Ã
f (x)
g(x)
!0
=
f 0 (x)g(x) − f (x)g 0 (x)
g 2 (x)
d
df dg
f (g(x)) =
·
dx
dg dx
d
df dg dh
f (g(h(x))) =
·
·
dx
dg dh dx
d
df dg dh du
f (g(h(u(x)))) =
·
·
·
dx
dg dh du dx
Power rule
(xα )0 = αxα−1
8