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. lim

 x

2


3 x
2


7 x

1



 b. lim x

1
8 x
2 x

3 x

5

1
2. Let ( )
 x x

1
. Find ( ) using the limit definition of the derivative.
3.

BC 1-2 Quiz #2 Name:_________________
Suppose y
  x
2
2
 x
for x

[0,6] and y (0)
 
3 . Using a step size of 2, construct the graph of the piecewise-defined function for y approx
using Euler’s method making sure to mark and label your endpoints clearly. y
6
3
2 4 6 x
-3
-6
4.
Given the graph of y

( ) below, sketch the graph of y
 
( ) .

BC 1-2
5. Refer to the graph of f

(not

) shown at the right to answer the following questions. a. For what values of x
(approximate) is
 decreasing? Justify your answer.
Quiz #2 Name:_________________
f
 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:_________________ f.
e.
c.
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.
x

3
 x

3
 d.
Lim f x x

3
( ( ))
Lim f x x

0
 g x x

3

( ( )

( )) y = g(x) f (0
 h )
 f (0) g.
Lim h

0 h h.
Lim x

1
 x

1


BC 1-2 Quiz #2 Name:_________________
`
Concepts:
7.
Evaluate the limit: lim x

0


  x
12


8.
Using the limit definition of the derivative where necessary, determine where
 x
4 
3 x
3 
3 x
2  x | is differentiable. That is, for which values of x .

BC 1-2 Quiz #2 Name:_________________
9.
Let ( )
 x
3 
3 x . 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 ( , ) such that there are two different tangent lines to f that go through ( , ) . Justify. b.
Describe the set of points, ( , ) , in the plane with the property that there are exatcly two different tangent lines to f that go through ( , ) . c.
Are there are there any points that have three tangent
We do all parts at once. Since the function f is odd, we look at the right half plane x

0 . We draw the line y
 
3 x (tangent at origin). Points in the fourth quadrant that are below this line will have only one tangent line passing through them. As examples, I’ve shown three such points.

BC 1-2 Quiz #2 Name:_________________
Now look at all the points above or on the line y
 
3 x .
These points will all have three tangent lines passing through them. We can see this by noting that the slope of tangent line is decreasing for x

0 and increasing for x

0 . Examples of points in this region are shown below,

BC 1-2 Quiz #2 Name:_________________
Finally, those points above the graph of f with x

0 have only one tangent line passing through them.