Name:
Math 1A Quiz #6
Section:
Each problem is worth 3 points. Show all your work to receive full credit.
1. Sketch the graphs of:
(a) a function on [−3, 2] which has a local maximum but no absolute maximum.
(b) a function on [−3, 2] which has a local maximum at which it is continuous but not differentiable.
(Endpoints are allowed to be local extrema.)
The first function cannot be continuous by the extreme value theorem. For the second, the standard example
is something like f (x) = −|x|.
2. Find the critical numbers of the function f (x) = x/(x3 + 1).
Recall that a critical number is a number in the domain of the function where the derivative is either zero or
undefined. We note first that the function is undefined when x3 + 1 = 0, which implies that x = −1. Now, we
compute the derivative itself:
f 0 (x) =
1 − 2x3
(x3 + 1) · 1 − x · (3x2 )
= 3
.
3
2
(x + 1)
(x + 1)2
p
This is zero at p
x = 3 1/2 and undefined at x = −1, but x = −1 isn’t in the domain. Thus the only critical
number is x = 3 1/2.
3. Find the linear approximation at a = 0 of the function g(x) = sinh2 (x) + ex , and use it to approximate
sinh2 (0.1) + e0.1 .
We first compute using the chain rule that g 0 (x) = 2·sinh(x)·cosh(x)+ex , so g 0 (0) = 2·sinh(0)·cosh(0)+e0 =
2 · 0 · 1 + 1 = 1. Since g(0) = sinh2 (0) + e0 = 02 + 1 = 1, the tangent line is y = x + 1. Thus, we obtain the
approximation sinh2 (0.1) + e0.1 = g(0.1) ≈ 1.1.