5
2.3.1 The Non-routine Test
1. Find values of
a
and
b
so that the line
2x+3y=a
is tangent to the graph of
2
bx x
f =
at the point where
x=3
. 2. Does
x
21
+ x
19
- x
-1
+ 2 = 0
have any roots between
–1
and ? Why or why not?
3. Let
fx=
ïî ï
í ì
+ +
≤ 1
, 1
1 ,
2
x x
bx x
ax
. Find
a
and
b
so that
f
is differentiable at
1
. 4. Find at least one solution to the equation
4x
3
– x
4
= 30
or explain why no such solution exists.
5. Is there an
a
such that
3 2
1 3
2 lim
2 2
3
− −
− −
+ −
→
x x
a x
ax x
x
exists? E xplain your answer.
2.3.2 The Routine Test
1. a What is the slope of the line tangent to
y=x
2
at
x=1
? b At what point does the tangent line touch the graph of
y=x
2
? 2. Find the slope of the line
x+3y=5.
3. If
x x
x f
+ =
5
, where is
f
increasing? 4. If
1 −
= x
x f
, find
x f
′
. 5. a Suppose
f
is a differentiable function. Does
f
have to be continuous? b Is
= x
f
î í
ì ≤
, 2
, x
x x
continuous? 6. Find the maximum value of
2
2 2
x x
x f
− +
− =
. 7. Find
1 1
lim
2 1
− −
→
x x
x
. 8. Do the indicated division:
1 1
2 3
− +
− −
x x
x x
. 9. If 5 is a root of
= x
f
, at what point if any does the graph of
x f
y =
cross the
x-
axis? 10. Consider
ïî ï
í ì
+ ≤
= 1
, 3
1 ,
2
x x
x x
x f
. a
Find
lim
1
x f
x
+
→
. b
What is the derivative of
x f
from the left at
x=1
sometimes called the left-hand derivative?
6
As soon as the non-routine tests were collected, the students were given the two-page routine test. They were told answers without explanations would be acceptable, but they
could show their work if they wished. Most students worked quickly, taking from 12 to 17.5 minutes to complete their routine exercises. None stayed the allotted half hour. E ach
question was assigned 10 points. The highest score was 100 out of 100 and the lowest score was 50, as compared to a high score of 90 and a low score of 53 for the AB calculus
students [13]. The mean score on the routine test was 75.3.
E ach problem on the non-routine test could be solved using a combination of basic calculus and algebra skills. The correspondence between routine questions and non-routine
problems is given in Table 3.
Non-routine Problem: 1 2 3 4 5
Corresponding Routine Questions: 1, 2
3, 4, 9 5, 10
6,9 7, 8
T
ABLE
3. Correspondence between routine questions and non-routine problems
3. The Results
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