ASSIGNMENT 4
There are two parts to this assignment. The first part is on WeBWorK — the link is available on the course
webpage. The second part consists of the questions on this page. You are expected to provide full solutions
with complete justifications. You will be graded on the mathematical, logical and grammatical coherence and
elegance of your solutions. Your solutions must be typed, with your name and student number at the top of
the first page. If your solutions are on multiple pages, the pages must be stapled together.
Your written assignment must be handed in before the start of your recitation on Friday, October
3. The online assignment will close at 9:00 a.m. on Friday, October 3.
1. Let f (x) =
(2x2 − 3)(x2 − 6)
. Find f 0 (6).
x2 + 2
2. Find the equations of all lines tangent to y =
x−1
x+1
which pass through the point (−1, 0).
3. Pascal’s triangle is the array of numbers whose first nine rows are shown below. Every row begins and
ends with “1”. Every other entry is equal to the sum of the two entries immediately above it. The top
row is called “row 0”, the next row is called “row 1”, and so on.
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
1
8 28 56 70 56 28 8
1
Let f and g be infinitely differentiable functions, and let Fn and Gn denote their nth derivatives, respectively. Using this notation, the first derivative of f (x)g(x) is equal to
f 0 (x)g(x) + f (x)g 0 (x) = F1 G0 + F0 G1 ,
and the second derivative of f (x)g(x) is equal to
f 00 (x)g(x) + 2f 0 (x)g 0 (x) + f (x)g 00 (x) = F2 G0 + 2F1 G1 + F0 G2 .
Note that the coefficients in each case are the entries in row 1 and row 2 of Pascal’s triangle.
(a) Assume that the nth derivative of f (x)g(x) is equal to
an Fn G0 + an−1 Fn−1 G1 + an−2 Fn−2 G2 + · · · + a2 F2 Gn−2 + a1 F1 Gn−1 + a0 F0 Gn ,
where the coefficients an , an−1 , an−2 , . . . , a2 , a1 , a0 are the entries in row n of Pascal’s triangle. Prove
that the (n + 1)st derivative of f (x)g(x) is then equal to
bn+1 Fn+1 G0 + bn Fn G1 + bn−1 Fn−1 G2 + · · · + b2 F2 Gn−1 + b1 F1 Gn + b0 F0 Gn+1 ,
where the coefficients bn+1 , bn , bn−1 , . . . , b2 , b1 , b0 are the entries in row n + 1 of Pascal’s triangle.
(b) Explain why you may conclude that the 100th derivative of f (x)g(x) is equal to
a100 F100 G0 + a99 F99 G1 + a98 F98 G2 + · · · + a2 F2 G98 + a1 F1 G99 + a0 F0 G100 ,
where the coefficients a100 , a99 , a98 , . . . , a2 , a1 , a0 are the entries in row 100 of Pascal’s triangle.