MATH 140
Sample Test 3
Name:
Fall 2023
PSU ID (LAST FOUR DIGITS) :
Instructions: Write clearly. Show all your work. Partial credit may be given.
1. Find the critical number of the function f (x) =
x2
.
x−1
2. Find the absolute maximum, M, and the absolute minimum, m, of f (x) = 4x3 − 15x2 + 12x on
the interval [−1, 1].
3. Determine the interval(s) on which the function f (x) = |x2 − 1| is increasing.
4
1
4. Determine the interval(s) on which the graph of the function f (x) = x 3 + 4x 3 is concave upward.
5. Find the asymptotes of the function f (x) =
1+x2
.
(3x+1)2
6. Find an equation of the slant asymptote of the graph of f (x) =
2x2 +x+1
x+1
7. Given
x2
f (x) = + ,
x 9
18x
f (x) = 2
,
(x + 9)2
0
54(3 − x2 )
f (x) =
(x2 + 9)3
00
i Domain of f
ii Vertical asymptote(s)
iii Horizontal asymptote
iv Interval(s) where f is increasing
v Interval(s) where f is decreasing
vi Local minimum point(s), (x,y)
vii Local maximum point(s), (x,y)
viii Interval(s) where f is concave up
ix Interval(s) where f is concave down
x x-coordinate(s) of inflection point(s) for f
xi Sketch the graph of f. Label all asymptotes, local maximum and minimum points, and
inflection points.
8. Find the coordinates of the point on the line 2x + y = 2 closest to the origin (0, 0).