BC 1-2
Problem Set #2
Spring 2012
Name
Due Date: Thursday, 9 Jan. (at beginning of class)
Please show appropriate work – no big calculator leaps – except as indicated. Work should be shown
clearly, using correct mathematical notation. Please show enough work on all problems (unless
specified otherwise) so that others could follow your work and do a similar problem without help.
Collaboration is encouraged, but in the end, the work should be your own.
1.
Suppose f ( x)  x n with n a negative integer. Use the limit definition of the derivative to prove that
f ( x)  nx n1 .
2.
Using the limit definition of the derivative where necessary, show that the function y  x | x |3 is
differentiable everywhere.
BC 1-2
Problem Set #2
Spring 2012
Name
Due Date: Thursday, 9 Jan. (at beginning of class)
3.
Find the value of c for which the parabola y  x 2  c is tangent to y  ln x . Use the fact that the
1
derivative of y  ln x is y  . Note: two functions f and g are tangent at x  a if and only if
x
f (a)  g (a) and f (a)  g (a) .
4.
Suppose that f is a function which is defined on all the reals and is differentiable everywhere.
a.
Evaluate lim
h0
f ( x  h 2 )  f ( x)
.
h
BC 1-2
Problem Set #2
Spring 2012
Name
Due Date: Thursday, 9 Jan. (at beginning of class)
4 (continued) Suppose that f is a function which is defined on all the reals and is differentiable
everywhere.
b. Suppose that g is a continuous function with lim  g (h)  0 . What conditions on g might
h0
f ( x  g (h))  f ( x)
f ( x  g (h))  f ( x)
 0 ? ii. lim
 f ( x)
guarantee that i. lim
h 0
h 0
h
h
[You need not prove your assertions, but justify any conjectures as fully as you are able.]