Math 105 Test 2 (75 points)
Name:
•Check that you have 9 questions on four pages.
•Show all your work to receive full credit for a problem.
You may use any of the following facts:
(uv)0=u0v+uv0u
v0
=vu0−uv0
v2(f◦g)0(x)=f0(g(x)) g0(x)
(bx)0= (ln b)bx(logbx)0=1
(ln b)x
(sin x)0= cos x(cos x)0=−sin x(tan x)0= sec2x
(arcsin x)0=1
√1−x2(arccos x)0=−1
√1−x2(arctan x)0=1
1+x2
1. (6 points) The Intermediate Value Theorem (IVT) says that if a function fis continuous on
the interval [a, b] and yis any number between f(a) and f(b), then f(c)=yfor some input c
in [a, b].
Let f(x)=2−x+3x4. Does the IVT guarantee that f(x) has a root in the interval [−1,1]?
Explain.
