Math 151 Exam 1 Review
J. Lewis
1. Write a vector equation for the line
a) through P(2,-6) and Q(4,-2).
b) y = 5x - 6
c) orthogonal to the line 4x + 3y = 10.
d) the horizontal line y=c for c a constant.
2. Find the distance from P(1, -3) to the line y=3x+2.

3. v  1,  6
Find

w  2, 3


a) the component of w in the direction of v .


b) the projection of w in the direction of v .


c) the cosine of the angle between v and w .
4. A force of 5N in the direction of N30  E is used to move an object 4m east and 5 m
north. Find the work done.
5. Determine whether the lines are parallel, perpendicular or neither. If not parallel, find
their intersection point and the smaller angle of intersection.



a)
r1 ( t )  ( 2  3 t ) i  (1  t ) j
b)
r1 ( t )  2 ,  6  t 3, 12
c)
r1 ( t )  1, 3  t 1,  4





r2 ( s )  (1  6 s ) i  (  1  2 s ) j

r2 ( s )  3 , 1  s 4 ,  1

r2 ( s )  10 ,  3  s  4 , 1
6.
 x2  5x  6

2
 x 9
f (x)  

1


x  3 and x   3
x  3 or x   3
a) For what value(s) of c does lim f ( x ) not exist?
x c
b) At what value(s) of c is f not continuous? For each value of c, what fails in the
definition of continuity and what is seen in the graph?
7.
 x2  7x

x
f (x)  
 4 x
  2 x  9
x0
0 x3
3 x
a) Show that f continuous on the whole real line.
b) At what value(s) of c is f not differentiable? For this/these value(s) of c, find
f '  ( c ) and f '  ( c ).
8. Evaluate each limit.
a)
1
lim x cos  
x 0
x
b)
1
lim cos  
x 0
x
2
d)
lim
x 2
1
lim cos  
x 
x
2
x  2x

c)
e)
2
x 4
lim
x 2
 6 | x |  18

x9

9. f ( x )  

 9 

 cos 
 x 

x  2x

2
x 4
2
f)
x  2x
lim
x  2
2
x 4
x9
9 x
Evaluate each limit.
a ) lim
x 9

f (x)
b ) lim
x 9
10. Evaluate each limit.

f (x)
c ) lim
x  
f ( x)
d ) lim f ( x )
x 
2
a)
c)
lim
x 
lim 
x 
2
9x  7x  3
b)
2x  4
x  4 x  x 

2
9x  7x  3
lim
2x  4
x  
lim 
d)
x   
x  4 x  x 

2
11. f ( x )  x  x  12
Find a positive integer a for which f(x) has a root in (a, a+1).
4
3
12. Find an equation of the tangent line to the given function at the given value of a.
i)
f ( x)  x
a 8
ii )
g ( x )  2 x  2 ( x  1)
2 3
3

iii )

r (t )  t t  3 i 
1
t
2
2
a  2

j
Find a vector equation and the slope - int equation.
at t  1
13. For each function, find the x-values at which the tangent line is horizontal.
a)
f ( x) 
x
( x  2)
2
b)
f ( x) 
x
x 1
c)
f ( x )  ( x  2 ) ( x  3)
2
3