BC 1-2
Quiz #2
Name:_________________
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Skills:
1.
Find each limit algebraically. Clearly show how you arrived at your answers. An answer without
any algebraic justification will receive little or no credit.
a.
 3

 x  7  1
lim 

x2 
2 x 


b.
8 x 2  3x  5
x 1
x 1
lim
2. Let f ( x) 
x
. Find f ( x ) using the limit definition of the derivative.
x 1
BC 1-2
3.
Quiz #2
Name:_________________
x2
for x  [0,6] and y (0)  3 . Using a step size of 2, construct the graph of the
2 x
piecewise-defined function for yapprox using Euler’s method making sure to mark and label your
Suppose y 
endpoints clearly.
y
6
3
2
-3
-6
4.
Given the graph of y  f ( x) below, sketch the graph of y  f ( x) .
4
6
x
BC 1-2
5.
Quiz #2
Refer to the graph of f  (not )
shown at the right to answer the
following questions.
Name:_________________
Graph of f 
a.
For what values of x
(approximate) is 
decreasing? Justify your
answer.
b.
For what value(s) of x (approximate) does  have a local minimum? Justify your answer.
c.
For what values of x (approximate) is  concave up? Justify your answer.
d.
For what value(s) of x (approximate) does  have inflection points? Justify your answer.
e.
If f (3)  2 , write the equation of the tangent line to the graph of f at x  3 .
f.
Sketch a graph of  on the same axes as the graph of .
BC 1-2
Quiz #2
Name:_________________
6. The graphs of f and g are shown at right. Evaluate the following: No work or explanation required. Be
Careful – look at the correct graph. (Write DNE if the limit does not exist).
y
= f(x)
a. Lim( f ( x))
x 0
b. Lim ( f ( x))
x 3
c. Lim ( f ( x))
x 3
d. Lim( f ( x))
x 3
e. Lim( f ( x)  g ( x))
x 0
f.
Lim ( f ( x)  g ( x))
x 3
y=g(x)
 f (0  h)  f (0) 
g. Lim 

h 0
h


 f ( x) 
h. Lim 

x 1
 x  1
BC 1-2
Quiz #2
Name:_________________
Concepts:
1  cos  x 6  

7. Evaluate the limit: lim 
x 0
x12


8. Using the limit definition of the derivative where necessary, determine where
f ( x) | x 4  3x3  3x 2  x | is differentiable. That is, for which values of x is f differentiable.
BC 1-2
Quiz #2
Name:_________________
9. Let f ( x)  x3  3x .
a. We saw in class that there is one line that is tangent to the graph of f that goes
through (0,1) . Find a point (a, b) such that there are two different tangent lines to f that
go through (a, b) . Justify.
b. Describe the set of points, (a, b) , in the plane with the property that there are exactly two
different tangent lines to f that go through (a, b) .
c. Are there are there any points that have three tangent