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Date
Apr 3, 2024
Type
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Uploaded by ProfessorFreedom13857
Calculus AB Assignment
More Exploration
1
Copyright © 2021 Apex Learning. See Terms of Use for further information. Images of the TI-84 calculator are used with the permission of Texas Instruments Incorporated. Copyright © 2011 Texas Instruments Incorporated.
1.
Below is a graph of the function y = f ( x )
.
6
5
4
3
2
1
-4
-2
0
-1
2
4
x
6
8
From your observations of this graph, state, with reasons, the values of x at which f is not differentiable.
As we know that F(x) is not differentiable at the point where x is discontinuous or there exist a
corner point. As we can see from the given graph that, there exist corner points at x=2, x=4,
and f(x) is discontinuous at x=6
Hence f(x) is not differentiable at x=2, x=4, x=6
2.
Consider the function f
(
x = x + |
x
| . Use graphical methods to discuss the differentiability of the function. In particular, find all values of x for which the function appears non-differentiable and explain why. Include at least one graph illustrating each point of non-differentiability.
3.
Consider the function f
(
x
= 1 —
x
4 — 5
x
2 + 4 . Use graphical methods to discuss the
differentiability of the function. In particular, find all values of x for which the function appears non-differentiable and explain why. Include at least one graph illustrating each point of
non-differentiability.
10
8
6
4
2
-4
-2
0
-2
2
x
4
-4
Calculus AB Assignment
More Exploration
Copyright © 2021 Apex Learning. See Terms of Use for further information. Images of the TI-84 calculator are used with the permission of Texas Instruments Incorporated. Copyright © 2011 Texas Instruments Incorporated.
2
x
2 – 2
x + 1
if
x ≤ 1
4.
Consider the function f (
x
=
2
x 3 –
3
x 2 – 3
x
if
1 < x < 4
1
x 4 + x
2 + 9
x – 24
if
4 ≤ x
4
Discuss the continuity and differentiability of the function at x = 1 and x = 4
. Prove your assertions mathematically (a graphic proof will NOT suffice).
5.
Below is the graph of y = g
'
(
x
) .
Is it possible, impossible, or certain that the function g is continuous at x = 2
? Explain. Is
it possible, impossible, or certain that the function g is differentiable at x = 2
? Explain.
2
5
4
3
2
1
-4
-2
0
2 x4
-1
Calculus AB Assignment
More Exploration
Copyright © 2021 Apex Learning. See Terms of Use for further information. Images of the TI-84 calculator are used with the permission of Texas Instruments Incorporated. Copyright © 2011 Texas Instruments Incorporated.
6.
Below is the graph of y = g
'
(
x
) .
[Note: The open circles at x = 2 are to imply that the function g
' is not defined there.] Answer the following questions about the function g
(
x
) .
A.
Is is possible, impossible, or certain that g is continuous at x = 2
? Explain.
B.
Is it possible, impossible, or certain that g
(
2)
=
0
? Explain.
C.
Is it possible, impossible, or certain that g has a vertical asymptote at x = 2
? Explain.
D.
Is it possible, impossible, or certain that g has a jump discontinuity at x = 2
?
E.
Is it possible, impossible, or certain that g has a removable discontinuity at x = 2
?
F.
Make a sketch of what the graph of y = g
(
x
)
could look like.
3
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