MthT 430 Chapter 9 Derivatives

Definition. (p. 149) The function f is differentiable at aif

lim h→ 0  f(a + h) − f(a) h = f′(a) exists.

N.B. The function f′ is a function whose domain is the set of all numbers a such that f′(a) exists.

The tangent line to the graph of f at (a,f(a)) is the line through (a,f(a)) with slope f′(a). By the point-slope form, the equation (formula) for the tangent line [to the graph of f at (a,f(a))] is

y = Ta(x) = f(a) + f′(a) (x −a).

If f is differentiable at a, the error of the tangent line approximation is

ϕa(x) ≡ f(x) − Ta(x), = f(x) − (f(a) + f′(a) (x −a) ).

Note that

lim x → a  ϕa(x) x − a = 0.

Theorem 1. If f is differentiable at a, then f is continuous at a.

Proof.

lim h → 0  (f(a+h) − f(a)) = lim h→ 0  f(a + h) − f(a) h ·h = f′(a)·0 = 0.

Examples

1.

f(x) = |x|. f is continuous (at all points in its domain).

f′(a) = ⎧ ⎪ ⎨ ⎪ ⎩ 1, a > 0, −1,   a < 0, undefined, a = 0.

2.

There are functions f which are continuous everywhere, but differentiable nowhere.

3.

For n = 1, 2, …, the power-n function , Power_n(x) = xⁿ, is differentiable (everywhere) and

Powern′(x) = n xn −1.

4.

We will assume that the trigonometric functions, sin and cos, are differentiable and

sin′(x) = cos(x), cos′(x) = − sin(x).

5.

Let

f(x) = ⎧ ⎪ ⎨ ⎪ ⎩ x sin(1/x), x ≠ 0, 0, x = 0.

Then f is continuous at 0, but not differentiable at 0. The difference quotient at 0 is

f(0 + h) − f(0) h = h sin(1/h) h = sin(1/h),

which does not have a limit as h → 0.

6.

Let

f(x) = ⎧ ⎪ ⎨ ⎪ ⎩ x2 sin(1/x), x ≠ 0, 0, x = 0.

Then f is continuous at 0, and differentiable at 0. The difference quotient at 0 is

f(0 + h) − f(0) h = h2 sin(1/h) h = h sin(1/h) → 0 as h → 0.

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On 18 Sep 2014, 21:29.