Calculus I Exam #1
Fall 2005
Name _____________________________
Show all your work on this paper. Solutions without correct supporting work will not be accepted. All
solutions must be exact unless stated otherwise in the problem.
1. Use the graph of y  f (x ) given below to find the limits indicated.
(2 points each)
a. lim f ( x)  _______________
x2
b. lim f ( x)  _______________
x  1
c. lim f ( x)  _______________
x 4
2. Evaluate without using a graph or table. (i.e., show that you worked it analytically)
a. lim
x 5
c. lim
 0
x 2  x  3 = ________________
cos  tan 

= ________________
(5 pts each)
3x 3  3x
= ________________________
x  1 2 x 2  2 x
b. lim
d. lim
x2
1 x
= ____________________________
2 x
3. Find all points of discontinuity. Also state which part of the definition of continuity fails at that particular
point. If there are none make sure you write ‘NONE” in the blank.
(3 points each)
x 1
a. f ( x) 
Discontinuous at x = _________ Reason: ________________________
x3
Removable or Nonremovable? ________________________
4  x , x  5
b. f ( x)  
2 x  11, x  5
Discontinuous at x = _________ Reason:__________________________
Removable or Nonremovable? ________________________
4. Find f (x) using the definition of the derivative:
f ( x)  3 x 2  x  5
5. Find the derivative. Do not simplify your answers. Circle your answers.
a.
y  4x3  3 x5 
b.
f ( x)  ln(sin x)
c. g ( x) 
d.
x 5  9e x
x2  3
y  x 3 4x 2  9
7
 sec x
x
(8 points)
(7 points each)
6. Write the equation of the tangent line to the graph of f ( x)  arctan x at the point where x  1. Write your
answer in slope-intercept form. (8 points)
7. Find
dy
: x 3 y 2  y  4x  5
dx
(8 points)
8. To estimate the height of a building, a stone is dropped from the top of the building into a pool of water at
ground level. How high is the building (in meters) if the splash is seen 6.8 seconds after the stone is
dropped? Recall that the position function is given by s (t )  4.9t 2  v0 t  s 0 .
(8 points)
9. Find
dy
:
dx
y  ( x  3) x
(8 points)