Math 151 Exam 1 Review
J. Lewis
1. Given two points, P(2, -1) and Q(1, 4) ,
a) Find a vector equation of the line through P and Q.
b) Find a vector orthogonal to this line. Find the general form of the line through P and Q.
c) Find the distance from R(3,7) to the line.
2. Given the points A(-1,2), B(2,1) and C(0,5), find angle ACB.
3. Let a=-2i+3j and b=<1, 2>. Find the vector projection of b onto a.
4. A 10 kg suitcase sits at the top of the ramp of a cruise ship which is 4 meters tall and has a horizontal
base of 2 meters. Assuming no friction, find the work done by gravity when the suitcase slides from the
top of the ramp to the bottom.
5. Find parametric equations for the line passing through the points (2, -3) and (1, 2) so that x(0)=2 and
y(0)=-3.
6. Describe the motion of a particle if the position of the particle is given by x=3sin(t) and y= -2cos(t),
0t  .
7. Two lines are given parametrically by
l 1: x1 (t )  4t  1 y1 (t )  t  2
l 2 : x2 ( s )  5 s  3
y2 ( s )  2 s
Find the pt of intersection and the smaller angle of intersection.
8. Compute each limit or show it does not exist:
x4  1
a) lim
x 1 x 2  1
b) lim
1 1

c) lim x 5
x 5 x  5
d) lim x sin( )  5
e) lim
x 1
x 2  2x  3
x 1
x3  x
x 1 x 2  5 x  4
x 0
1
x
 2
5  5 x if

if
9. Given f ( x)   3
 9  x if


x5
x5
Determine whether f is continuous at x=5 from the left, from
x5
the right, both or neither.
10. Find all discontinuities of f(x). Give a reason from the definition of continuity and a graphical reason
for each discontinuity, a, and determine whether f is left continuous at a, right continuous at a or
neither.
 4 x 8

x4

f ( x)  
1
2
 x  16
 2
 x  10 x  24
0 x4
x4
4 x
11. Find the values of m and b which make f(x) continuous and differentiable at x = `1.
3 x 2  2 x if

f ( x)   mx  b if


x 1
1 x
12. Find an integer N so that x 3  5  x has a solution in the interval [N, N+1].
13. Compute the following limits:
a)
b)
6x 2  x  3
lim
x  2  5 x  3 x 2
lim
x 
9x 2  4
x2
c)

lim x  x 2  3x  2
x 

14. The position of a particle moving along the x axis is given by the function s(t )  t 3  t  1 , where s is
in meters and t is in seconds.
a) Compute the average velocity of the particle over the interval [2,3].
b) Compute the instantaneous velocity at t=2. Find the slope intercept equation of the tangent line
at t=2.



15. Given the curve r (t )  (t 2  t  2) i  4t 2 j , find a vector tangent to the curve and find parametric
equations for the line tangent to the curve at the point (-2,4).
16. Write the limit definition of the derivative for the function
17.a) Use the limit definition to find the derivative of
f ( x) 
f ( x)  5 x 2 
1
 9 at a=2.
x
1
.
x2
4( x  h) 2  2 x  h  4 x 2  2 x
b) Evaluate lim
. Use anything you know.
h 0
h
18. Find all values of a for which f is not differentiable at a.
a)
f ( x) | x |
c)
f ( x)  x
23
b)
f ( x) | x 2  4 |
d)
2 x 2  5 x
f ( x)  
6

x 1
x 1
19. For each function, determine all x-values at which f is not differentiable. What is seen in the graphs
of f and of f ‘ at these points?
a)
 x1 3 x  1
f ( x)  
2 x  1 1  x
b)
 x1 3 x  1
f ( x)   x  2
1 x
 3
20. Find the average rate of change of 𝑓(𝑥) = 2𝑥 + √𝑥 from 𝑥 = 4 𝑡𝑜 𝑥 = 9.