i.
f
(
x
) =
x
3

19
x
2
+
x
+ 1
x
2

2
x

1
ii.
f
(
x
) =
(
x
2
+ 1
x <

1 or
x >
1
x

1
≤
x
≤
1
iii.
f
(
x
) =
(
x
3

7
x
x <
0
0
x >
0
iv.
f
(
x
) =
√
x
+ 7
v.
f
(
x
) =
x
2

25
x
+ 5
2. Find a value of
c
for each of the following functions making the function continuous:
i.
f
(
x
) =
(

x
 
1
x
≤
2
x
2
+
c
x >
2
ii.
f
(
x
) =
(
x

1
x

2
x
≤ 
1
cx
2
+ 1
x >

1
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The Derivative:
•
Know the definition of the derivative and be able to use it to calculate derivatives
•
Be able to find equations for tangent lines at points
•
Know what it means to be differentiable
•
Understand relationship between slopes of secants and tangents and average and in
stantaneous rates of change
Exercises:
1. Consider the function
f
(
x
) =
x
2
+
x
+ 1.
i. Give the equation of the secant line through the points (1
, f
(1)) and (2
, f
(2)) and
give its slope.
ii. Using the definition of the derivative, compute
f
0
(1)
iii. Using the definition of the derivative, compute
f
0
(
x
)
2. For the following give equations for the lines tangent to the given point
i.
f
(
x
) =
1
x
+ 1
,
x
= 1.
ii.
f
(
x
) =
x
2

1,
x
=

2.
iii.
f
(
x
) =
x
3
+ 1,
x
= 2
iv.
f
(
x
) =
√
x

1 +
x
,
x
= 2
3. Find the equation for the line perpendicular to
f
(
x
) =
x
2
+ 2
x
at
x
= 1.
4. Use linear approximations of
f
(
x
) =
√
x
near 1 and 4 to estimate
√
3.
Challenge Question:
Given a quadratic (degree 2) polynomial, there is no gaurantee
that there are any real roots. For example, the polynomial
x
2
+ 1 has no real roots. Can the
same be said for cubic (degree 3) polynomials?
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 Fall '11
 TA
 Polynomials, Quadratic Formula, Exponents, Derivative, Intermediate Value Theorem, Inequalities, Continuous function, lim P

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