Math 141 Homework #5
Due Tuesday, 9/18/07
Extra Problems
Problem #1 We know that every polynomial function f(x) can be written in the form
f(x) =
n
X
i=0
aixi
where n≥0 is an integer and an, an−1,...,a1, a0are real numbers.
Write down a formula for f0(x) in terms of the numbers an,an−1,...,a1,a0and the variable x. You should
express your answer using summation notation, but you don’t have to prove it.
Problem #2 Let f(x) and g(x) be functions, and let aand bbe constants. Using the definition of the
derivative of a function, write down a clear, well-organized proof of the fact that
d
dx (a·f(x) + b·g(x)) = a·f0(x) + b·g0(x).
You are encouraged to use the proofs of the Constant Multiple, Sum, and Difference Rules (textbook pp. 186–
187, or from class on Wednesday 9/12) as templates in constructing your own proof.
Problem #3 Construct a function q(x) with domain Rthat is differentiable but not second-differentiable.
That is, q0(x) is defined and continuous on R, but q00(x) has a discontinuity (say, at x= 0).
Bonus part: For all positive integers n, construct a function qsuch that qis (n−1)th -order differentiable
but not nth-order differentiable. That is,
q, dq
dx ,d2q
dx2, ..., dn−1q
dxn−1
are all defined and continuous, but dnq
dxnhas a discontinuity (again, say, at x= 0).