SCIENCE ONE: MATHEMATICS ASSIGNMENT 6 (Due Nov. 23, 2011)
There are two parts to this assignment. The first part is online and is due by 10:00 am on Wed. Nov. 23;
you will need to login at www.mathxl.com. The second part consists of the questions on this page and is also
due by 10:00 am on Wed. Nov. 23. You are expected to provide full solutions with complete arguments and
justifications. You will be graded primarily on the correctness, clarity and elegance of your solutions. Your
answers must be typeset or very neatly written. They must be stapled, with your name and student number
at the top of each page.
1. Use linear approximation and the remainder formula at the indicated point a to:
i. estimate the given value (using linear approximation);
ii. determine an upper bound on the absolute value of the error in part i.;
iii. determine a closed interval that is guaranteed to contain the exact value.
Give exact numerical expressions for parts i.–iii. Then
iv. verify your answer to part iii., using a (presumably) accurate decimal approximation of the exact value
obtained with a calculator.
p
√
(a) 9.05, a = 9,
(b) (3.9)5 , a = 4,
(c) sin 29◦ , a = 30◦
√
√
2. Which is more accurate: (a) using linear approximation for x at 4 to estimate 6.5, or (b) using linear
approximation at 9? (Remember to justify your answer correctly, clearly and elegantly.)
3. Let I be an interval, and let a be a point in I. Assume that if f is any function whose second derivative
f (2) exists on I, and x is any point in I not equal to a, then the remainder
R1 (x) = f (x) − f (a) − f (1) (a)(x − a)
in the first-order Taylor polynomial (or linear approximation) for f centred at a, satisfies
R1 (x) =
f (2) (c)
(x − a)2
2
for some (i.e. at least one) point c between a and x (actually, this was proved in class).
Now suppose f is any function whose third derivative f (3) exists on I, let x be a point in I not equal to
a, and for each t in I, let
R2 (t) = f (t) − f (a) − f (1) (a)(t − a) −
f (2) (a)
(t − a)2
2
be the remainder in the second-order Taylor polynomial for f centred at a.
(a) Assuming a < x:
i. use the Generalized Mean Value Theorem, with the function R2 (t) and another suitable function
of t defined on I, to prove that there exists some (i.e. at least one) number d strictly between a
and x such that
R20 (d)
R2 (x) =
(x − a)3
3(d − a)2
(you may use ideas from the induction argument used in class for a general Rk (x), but use the
GMVT for this specific case directly as instructed here); then
ii. prove that there exists some number c between a and d such that
R20 (d) =
f (3) (c)
(d − a)2 ,
2
R2 (x) =
f (3) (c)
(x − a)3 .
3!
and therefore
(b) Assuming x < a, repeat parts i. and ii.
(Now you could use the same method and the last formula for R2 (x) to prove the analogous formula for
R3 (x), then use that result to prove the formula for R4 (x), etc. The idea of induction makes precise what
is meant by “etc.” and the point of this question is to make more concrete the ideas used in the induction
argument done in class, for a specific k and k + 1.)