Math 151 Week in Review for sections 2.5-2.7
J. Lewis
1.
Find the value of m lim f ( x ) exists.
x a
a)
b)
 mx  4
f ( x)  
 3x  7
x2
a=2
2 x
 x 2  x  20

f ( x )   x 2  16

m

x   4, x  4
a4
x4
2. Find all discontinuities of each function and give a reason for each discontinuity.
x2  5x
a)
f ( x) 
b)
 x2  5x
 2
 x  25
g ( x)  
 x


x 2  25
Which of these is removable?
x2
2 x
c)
  1 
cos
x 1
f ( x )    x  1 
 x  1
1 x
d)

 1 
( x  1) cos 
 x 1
f ( x)  
 x 1

( x  1)1 3
1 x

3. Find the value of A for which the function is continuous at x=2.
 6 x  12
 x2  4

f ( x)  
 A(3  x ) 2 3


x2
2 x
At what other x-value is f(x) discontinuous?
4. Show that c  1 
2
c  4 for some number c between 0 and 2.
5. Evaluate each limit.
a)
c)
e)
4 x 4  25 x 3
lim
b)
x
25 x  30 x
lim
16 x 2  10 x
x
x  
lim
x

4
x2  8x  x
d)
lim
x
lim
x  
16 x 2  10 x
x
7 x 3  9 x  15
x2  1

6. Find the equation of the tangent line to the curve y  4 x  3 x
at a  2 .
2
7. An object travels along the path given parametrically by x (t )  t
a) Find a vector equation of the tangent line to the path at t=3sec.
2
y ( t )  4t  2 .
b) Find the slope intercept equation, y=mx+b , of this tangent line.
c) A second object travels along the path with vector equation

r (s) 

s
i  s j . Find the coordinates of the point where the paths intersect.
4
What is the angle of intersection?
d) If t=0 is noon and s=0 is 10 am, do the objects collide? (s and t are in hours)