Math 2250 Written HW #5 Solutions
1. Shown below is the graph of the function p(x) = x2 − 3x along with the tangent line at the
point (3, 0). What is the equation of this tangent line?
4
3
2
1
-1
1
2
3
4
-1
-2
-3
-4
Answer: First we want to find the slope of the tangent line, which of course is the derivative
of p at x = 3:
p(3 + h) − p(3)
h→0
h
(3 + h)2 − 3(3 + h) − 32 − 3(3)
= lim
h→0
h
2
9 + 6h + h − 9 − 3h − 0
= lim
h→0
h
3h + h2
= lim
h→0
h
= lim (3 + h)
p0 (x) = lim
h→0
= 3.
Therefore, the tangent line we want is the line of slope 3 passing through the point (3, 0).
Using the point-slope formula, this is the line
y − 0 = 3(x − 3)
y = 3x − 9.
2. (a) Give an example of two functions f (x) and g(x) so that f and g are both continuous but
not differentiable at x = 0, but the function h(x) = f (x)g(x) is differentiable at x = 0.
Answer: It turns out that our favorite example of a function which is continuous but
not differentiable, the absolute value function, will work here.
Let f (x) = g(x) = |x|. Then
h(x) = f (x)g(x) = |x|2 = x2 ,
which is certainly differentiable at x = 0 since
(0 + h)2 − 02
h2
= lim
= lim h = 0.
h→0 h
h→0
h→0
h
lim
1
(b) Is it possible to find functions f (x) and g(x) so that f and g are continuous but not
differentiable at x = 0 and the function k(x) = f (x) + g(x) is differentiable at x = 0? If
it is possible, find two such functions. If it is not possible, explain why.
Answer: Yes. Again, we can let f (x) = |x|. We can cancel out the lack of differentiability by adding g(x) = −|x|. Then
k(x) = f (x) + g(x) = |x| + (−|x|) = 0,
which is certainly differentiable everywhere, including at x = 0.
2