The left-hand and right-hand derivatives off at a are defined by f ' ( a ) = lim h → 0 − f ( a + h ) − f ( a ) a and f ' + ( a ) = lim h → 0 + f ( a + h ) − f ( a ) h if these limits exist. Then f' ( a ) exists if and only if these one-sided derivatives exist and are equal. (a) Find f ' − ( 4 ) and f ' + ( 4 ) for the function f ( x ) = { 0 if x ≤ 0 5 − x if 0 < x < 4 1 5 − x if x ≥ 4 (b) Sketch the graph of f (c) Where is f discontinuous? (d) Where is f not differentiable ?
The left-hand and right-hand derivatives off at a are defined by f ' ( a ) = lim h → 0 − f ( a + h ) − f ( a ) a and f ' + ( a ) = lim h → 0 + f ( a + h ) − f ( a ) h if these limits exist. Then f' ( a ) exists if and only if these one-sided derivatives exist and are equal. (a) Find f ' − ( 4 ) and f ' + ( 4 ) for the function f ( x ) = { 0 if x ≤ 0 5 − x if 0 < x < 4 1 5 − x if x ≥ 4 (b) Sketch the graph of f (c) Where is f discontinuous? (d) Where is f not differentiable ?
Solution Summary: The author explains how to calculate the left-hand derivative of f at x=a.
The left-hand and right-hand derivatives off at a are defined by
f
'
(
a
)
=
lim
h
→
0
−
f
(
a
+
h
)
−
f
(
a
)
a
and
f
'
+
(
a
)
=
lim
h
→
0
+
f
(
a
+
h
)
−
f
(
a
)
h
if these limits exist. Then f'(a) exists if and only if these one-sided derivatives exist and are equal.
(a) Find
f
'
−
(
4
)
and
f
'
+
(
4
)
for the function
f
(
x
)
=
{
0
if
x
≤
0
5
−
x
if
0
<
x
<
4
1
5
−
x
if
x
≥
4
(b) Sketch the graph of f
(c) Where is f discontinuous?
(d) Where is f not differentiable?
With integration, one of the major concepts of calculus. Differentiation is the derivative or rate of change of a function with respect to the independent variable.
The left-hand and right-hand derivatives of f at a are defined by
f(a + h) – f(a)
f(a + h) – f(a)
f'_(a) =
lim
and f',(a) = lim
h→0
if these limits exist. Then f'(a) exists if and only if these one-sided derivatives exist and are equal.
Find f'_(0) and f'.(0) for the given function f. (If an answer does not exist, enter DNE.)
so
if x 0
f'_(0)
%D
f'4(0)
Is f differentiable at 0?
Yes
No
So
if x 0
f'_(0)
f'4(0)
Is f differentiable at 0?
Yes
No
O O
O O
f(r +h) – f(x)
Use the limit definition f'(x) = lim
to find the derivative of f(x) =
h
2x2 +1.
3) Use the limit definition of the derivative to find f' (x) if f(x) =-.
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Differential Equation | MIT 18.01SC Single Variable Calculus, Fall 2010; Author: MIT OpenCourseWare;https://www.youtube.com/watch?v=HaOHUfymsuk;License: Standard YouTube License, CC-BY