Left- and Right-Hand Derivatives The left-hand and right-hand derivatives of f at a are defined by f − ' ( a ) = lim h → 0 − f ( a + h ) − f ( a ) h 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 onesided derivatives exist and are equal. 64. Find f ′ − ( 0 ) and f + ′ ( 0 ) for the given function f . Is f differentiable at 0? (a) f ( x ) = 0 if x ⩽ 0 x if x > 0 (b) f ( x ) = 0 if x ⩽ 0 x 2 if x > 0
Left- and Right-Hand Derivatives The left-hand and right-hand derivatives of f at a are defined by f − ' ( a ) = lim h → 0 − f ( a + h ) − f ( a ) h 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 onesided derivatives exist and are equal. 64. Find f ′ − ( 0 ) and f + ′ ( 0 ) for the given function f . Is f differentiable at 0? (a) f ( x ) = 0 if x ⩽ 0 x if x > 0 (b) f ( x ) = 0 if x ⩽ 0 x 2 if x > 0
Solution Summary: The author explains that f is differentiable at x=a if left hand derivative equals to right-hand derivative.
Left- and Right-Hand Derivatives The left-hand and right-hand derivatives of
f
at
a
are defined by
f
−
'
(
a
)
=
lim
h
→
0
−
f
(
a
+
h
)
−
f
(
a
)
h
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 onesided derivatives exist and are equal.
64. Find
f
′
−
(
0
)
and
f
+
′
(
0
)
for the given function
f
. Is
f
differentiable at 0?
True or false. if false, correct statement. if true, explain why.
a. if a function, f, is continuous at a point c, then f is differentiable at the point c.
b. if a function is concave down on its domain, then it will have a relative maximum.
c. The derivative of a sum is the sum of its derivatives.
d. the derivative of a function, f(x), is equal to limh->0 (f(x+h)-f(x))/h for all x values in the domain
e. if f has an absolute minimum at c, then f'(c)=0
6. State the definition of the derivative of a function f. Use the definition of the derivative toevaluate limx→0ln(1 + x)xif it exists. If the limit does not exist, explain why.
Finding the Derivative by the Limit Process In the given question/s, find the derivative of the function by the limit process.:-
f (x) = 1 /x - 1
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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