Math 170010
Homework 9/6
Fall 2011
Goal:
•
Compute derivatives of all powers of
x
.
•
Notice the recurring pattern that results from factoring a diFerence of powers: you get “some
number of copies of some object raised to some power.”
Exercises:
NOTE: My directions strongly suggest doing these problems with
x
and
u
. If you prefer to use
x
and
x
+
h
, I don’t mind. Try to build tables like mine, but expect that your algebra will look
diFerent than mine.
1. Consider the function
f
(
x
) =
x
2
.
(a) Using ±xed end
x
and free end
u
write down the secant slope.
(b) ²actor this, aiming to cancel (
x

u
).
(c) Let
u
→
x
to get tangent slope.
(d) Assemble your answers in a table that looks like this:
²unction
Secant
²actored
Tangent
x
2
x
2

u
2
x

u
(
x

u
)(
x
+
u
)
x

u
2
x
2. Repeat for the function
f
(
x
) =
x
3
. Add another line to the table.
HINT: DiFerence of cubes factors as (
x
3

u
3
) = (
x

u
)(
x
2
+
ux
+
u
2
).
3. Repeat for the function
f
(
x
) =
x
4
. Add another line to the table.
HINT: DiFerence of fourth powers factors as (
x
4

u
4
) = (
x

u
)(
x
3
+
x
2
u
+
xu
2
+
u
3
).
4. Repeat for the function
f
(
x
) =
x
5
. You might be able to guess how to factor (
x
5

u
5
) now.
If not, google it. But then you have to try
x
6
on your own.
5. Look at your table and see if you can guess the answers to the following questions.
(a) What would the factored secant slope look like for
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 Fall '08
 STAFF
 Calculus, Factoring, Derivative, secant slope, Berlin UBahn, tangent slope, x−u, factored secant slope

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