4. (a) It is known that differentiation satisfies the four rules
(1) $\quad \frac { d } { d x } ($ constant $) = 0$,
(2) $\quad \frac { d } { d x } ( x ) = 1$,
(3) $\frac { d } { d x } ( a f ( x ) + b g ( x ) ) = a \frac { d f } { d x } + b \frac { d g } { d x }$ for any constants $a , b$, and
(4) $\frac { d } { d x } ( f ( x ) \cdot g ( x ) ) = \frac { d f } { d x } g ( x ) + f ( x ) \frac { d g } { d x }$. .From these rules alone show that, for $n = 1,2,3 , \ldots$,
$$\frac { d } { d x } \left( x ^ { n } \right) = n x ^ { n - 1 } .$$
Use Rule (4) to find the derivative of the function
$$f ( x ) = \frac { 1 } { x ^ { n } } .$$
(It will help you to notice that $x ^ { n } \cdot \frac { 1 } { x ^ { n } } = 1$.)
(b) A careless calculus student remembers Rules (1), (2) and (3) correctly, but thinks that Rule (4) says
$$\frac { d } { d x } ( f ( x ) \cdot g ( x ) ) = \frac { d f } { d x } + f ( x ) \frac { d g } { d x } .$$
What will he or she compute for
$$\frac { d } { d x } \left( x ^ { 4 } \right)$$
- A set of 12 rods, each 1 metre long, is arranged so that the rods form the edges of a cube. Two corners, $A$ and $B$, are picked with $A B$ the diagonal of a face of the cube.
An ant starts at $A$ and walks along the rods from one corner to the next, never changing direction while on any rod. The ant's goal is to reach the corner $B$. A path is any route taken by the ant in travelling from $A$ to $B$.
(a) What is the length of the shortest path, and how many such shortest paths are there?
(b) What are the possible lengths of paths, starting at $A$ and finishing at $B$, for which the ant does not visit any vertex more than once (including $A$ and $B$ )?
(c) How many different possible paths of greatest length are there in (b)?
(d) Can the ant travel from $A$ to $B$ by passing through every other vertex exactly twice before arriving at $B$ without revisiting $A$ ? Give brief reasons for your answer.
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