8.24
Using the A-method, find the derivative of
8.25
Show that a differentiable function f(x) is continuous.
8.27
Find the derivative of /(x) = x
1
'
3
.
So,
8.26
Show that the converse of Problem 8.25 is false.
Consider the function /(x) = |x| at x = 0. Clearly, / is continuous everywhere. However,
When
and, when
Therefore,
does not exist.
8.28
8.29
Find the point(s) at which the tangent line to the parabola y = ax
2 + bx + c is horizontal. (Notice that the
solution to this problem locates the "nose" of the parabola.)
y' = 2ax + b is the slope of the tangent line. A line is horizontal if and only if its slope is 0. Therefore, we
must solve 2ax + b = 0. The solution is x=—b/2a. The corresponding value of y is (4ac — b
2 )/4a.
Let f(x) be a function with the property that /(« + v) = f(u)f(v) for all u and v, and such that /(O) = /'(O) =
1. Show that /'(*)=/(*) for all*.
= /W/'(0)=/W'l=/(x)
So,
52
CHAPTER 8
Hence,
So.
/is continuous at x.
So,
Thus,m