MATH S1101X(3). MIDTERM EXAM 1. PAGE 1 OF 10. COLUMBIA UNIVERSITY.
CALCULUS I (MATH S1101X(3)). 2ND SAMPLE MIDTERM 1 – JUNE 14, 2012.
COLUMBIA UNIVERSITY CALCULUS I (MATH S1101X(3)) 2ND SAMPLE MIDTERM 1 – JUNE 14, 2012 INSTRUCTOR: DR. SANDRO FUSCO
FAMILY NAME: ______________________________________________________________ GIVEN NAME: ______________________________________________________________
INSTRUCTIONS:
1. 2. 3. 4. 5.
Answer all nine (9) questions. Your work must justify the answer you give. Point values are as shown. No calculators, lecture notes and/or books are permitted. This is the first of ten (10) pages.
MATH S1101X(3)
Question
Points
1
10
2
10
3
10
4
10
5
10
6
10
7
10
8
10
9
20
Total
100
MIDTERM EXAM 1
Marks
PAGE 1 OF 10
Problem 1: (10 Points) Find the domain and sketch the graph of the function
MATH S1101X(3)
f ( x) =
MIDTERM EXAM 1
x x2
.
PAGE 2 OF 10
Problem 2: (10 Points) a) How is the graph of y
= f ( x ) related to the graph of y = f ( x ) ?
b) Sketch the graph of
y = sin x
c) Sketch the graph of
y=
MATH S1101X(3)
x
MIDTERM EXAM 1
PAGE 3 OF 10
Problem 3: (10 Points) Let f ( x) = 3 − x and state their domains.
MATH S1101X(3)
g ( x) = x 2 − 1 . Find the functions f + g , f − g , f ⋅ g , f g and
MIDTERM EXAM 1
PAGE 4 OF 10
Problem 4: (10 Points) Starting with the graph of y = ex, write the equation of the graph that results from a. Reflecting about the line
y = 4.
b. Reflecting about the line x = 2 .
MATH S1101X(3)
MIDTERM EXAM 1
PAGE 5 OF 10
Problem 5: (10 Points) Express the given quantity as a single logarithm. a)
ln (a + b ) + ln(a − b ) − 2 ln (c )
b)
1 ln 1 + x 2 + ln ( x ) − ln (sin x ) 2
(
)
MATH S1101X(3)
MIDTERM EXAM 1
PAGE 6 OF 10
Problem 6: (10 Points) Sketch the graph of the following function and use it to determine the values of lim x→a f ( x ) exists.
2 − x f (x ) = x 2 ( x − 1)
MATH S1101X(3)
a for which
if x < −1 if − 1 ≤ x < 1 if x ≥ 1
MIDTERM EXAM 1
PAGE 7 OF 10
Problem 7: (10 Points) Evaluate the limit if it exists. If the limit does not exist, explain why.
a)
b)
1 1 − lim t →0 t 1+ t t
(
lim e −2 x ⋅ cos( x )
x → +∞
MATH S1101X(3)
)
[Hint: Use the Squeeze Theorem]
MIDTERM EXAM 1
PAGE 8 OF 10
Problem 8: (10 Points) Use the Intermediate Value Theorem to show that there is a root of the equation 2 x3 - 3 x2 - 1 = 0 in the interval (1, 2).
MATH S1101X(3)
MIDTERM EXAM 1
PAGE 9 OF 10
Problem 9: (20 Points) Find the values of
a and b that make f continuous everywhere. x 2 -4 x-2 f (x) = ax2 − bx + 3 2x − a + b
