Math 3210-1
HW 14
Due Tuesday, July 17
The Derivative
1. For each of the following functions defined on R, give the set of points at which it is not differentiable.
Sketches will be helpful. No proofs are required.
(a) e|x|
(b) sin |x|
(c) | sin x|
(d) |x| + |x − 1|
(e) |x2 − 1|
(f) |x3 − 8|
2. Let f (x) = x2 sin
1
for x 6= 0 and f (0) = 0.
x
(a) Use Theorems 81 and 82 to show that f is differentiable at each a 6= 0 and calculate f 0 (a). Use,
without proof, the fact that sin x is differentiable and that cos x is its derivative.
(b) Use the definition to show that f is differentiable at x = 0 and that f 0 (0) = 0.
(c) Show that f 0 is not continuous at x = 0.
1
3. Let f (x) = x sin
for x 6= 0, and f (0) = 0.
x
(a) Observe that f is continuous at x = 0.
(b) Is f differentiable at x = 0? Justify your answer.
4. Let f (x) = x2 for x rational and f (x) = 0 for x irrational.
(a) Prove that f is continuous at x = 0.
(b) Prove that f is discontinuous at all x 6= 0.
(c) Prove that f is differentiable at x = 0. Warning: You cannot simply claim f 0 (x) = 2x.
The Mean Value Theorem
5. Prove that | cos x − cos y| ≤ |x − y|.
6. Suppose that f is differentiable on R and that f (0) = 0, f (1) = 1 and f (2) = 1.
(a) Show that f 0 (x) =
0
(b) Show that f (x) =
1
2
1
7
for some x ∈ (0, 2).
for some x ∈ (0, 2).
7. Show that ex ≤ ex for all x ∈ R.
8. Show that sin x ≤ x for all x ≥ 0. Hint: Show that f (x) = x − sin x is increasing on [0, ∞).
9. Prove that if f and g are differentiable on R, if f (0) = g(0) and if f 0 (x) ≤ g 0 (x) for all x ∈ R, then
f (x) ≤ g(x) for x ≥ 0.