# ASSIGNMENT 1·12 for SECTION 001 ```ASSIGNMENT 1&middot;12 for SECTION 001
There are two parts to this assignment. The first part is online at www.mathxl.com. The second part
consists of the questions on this page. You are expected to provide full solutions with complete arguments
must be typeset or very neatly written. They must be stapled, with your name and student number at the top
of each page.
1. In this question you will use “mathematical induction” to prove the Generalized Product Rule: given n
0
differentiable functions f1 , f2 , f3 , . . . , fn , the derivative (f1 f2 f3 &middot; &middot; &middot; fn ) will have n terms, each of which
is the product f1 f2 f3 &middot; &middot; &middot; fn but with one of the factors replaced by its derivative. In mathematical
shorthand, the Rule is as follows:
0
(f1 f2 f3 &middot; &middot; &middot; fn ) = f10 f2 f3 &middot; &middot; &middot; fn + f1 f20 f3 &middot; &middot; &middot; fn + f1 f2 f30 &middot; &middot; &middot; fn + &middot; &middot; &middot; + f1 f2 f3 &middot; &middot; &middot; fn0 .
(a) Explain why the Rule is true in the case n = 2.
(b) Let k be a positive integer. Prove that if the Rule is true for n = k, then it is true for n = k + 1.
(c) Explain why you may conclude that the Rule holds for, for example, n = 1000.
2. Find all points on the curve y =
x2
1
where the tangent line is horizontal.
+x+1
3. Let f (x) = (x − a)m (x − b)n where m and n are positive integers. Prove that the curve y = f (x) has a
horizontal tangent line between x = a and x = b.
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