One-sided derivatives The right-sided and left-sided derivatives of a function at a point a are given by f + ′ ( a ) = lim h → 0 + f ( a + h ) − f ( a ) h a n d f − ′ ( a ) = lim h → 0 − f ( a + h ) − f ( a ) h , respectively, provided these limits exist. The derivative f ′( a ) exists if and only if f + ′( a ) = f − ′( a ) . a. Sketch the following functions. b. Compute f + ′( a ) and f − ′( a ) at the given point a. c. Is f continuous at a? Is f differentiable at a? 32. f ( x ) = { 4 − x 2 if x ≤ 1 2 x + 1 if x > 1 ; a = 1
One-sided derivatives The right-sided and left-sided derivatives of a function at a point a are given by f + ′ ( a ) = lim h → 0 + f ( a + h ) − f ( a ) h a n d f − ′ ( a ) = lim h → 0 − f ( a + h ) − f ( a ) h , respectively, provided these limits exist. The derivative f ′( a ) exists if and only if f + ′( a ) = f − ′( a ) . a. Sketch the following functions. b. Compute f + ′( a ) and f − ′( a ) at the given point a. c. Is f continuous at a? Is f differentiable at a? 32. f ( x ) = { 4 − x 2 if x ≤ 1 2 x + 1 if x > 1 ; a = 1
Solution Summary: The author illustrates the function f(x)=cc4-x
One-sided derivativesThe right-sided and left-sided derivatives of a function at a point a are given by
f
+
′
(
a
)
=
lim
h
→
0
+
f
(
a
+
h
)
−
f
(
a
)
h
a
n
d
f
−
′
(
a
)
=
lim
h
→
0
−
f
(
a
+
h
)
−
f
(
a
)
h
,
respectively, provided these limits exist. The derivative f′(a) exists if and only if f+′(a) = f−′(a).
a.Sketch the following functions.
b.Compute f+′(a) and f−′(a) at the given point a.
c.Is f continuous at a? Is f differentiable at a?
32.
f
(
x
)
=
{
4
−
x
2
if
x
≤
1
2
x
+
1
if
x
>
1
;
a
=
1
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.
Only 100% sure experts solve it correct complete solutions ok
rmine the immediate settlement for points A and B shown in
figure below knowing that Aq,-200kN/m², E-20000kN/m², u=0.5, Depth
of foundation (DF-0), thickness of layer below footing (H)=20m.
4m
B
2m
2m
A
2m
+
2m
4m
sy = f(x)
+
+
+
+
+
+
+
+
+
X
3
4
5
7
8
9
The function of shown in the figure is continuous on the closed interval [0, 9] and differentiable on the open
interval (0, 9). Which of the following points satisfies conclusions of both the Intermediate Value Theorem
and the Mean Value Theorem for f on the closed interval [0, 9] ?
(A
A
B
B
C
D
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