Linearity in the function: For any h>0, s,ter, any f,g=C(R,R), any YER ¹+1, any wer n+1 and any XER, we have D[sf + tg, x; h, y, w] = sD[f, x; h, y, w] + tD[g, x; h, y, w]. Product rule: For any h>0, any f,geC∞ (R,R), any YER' n+1 n+1 any WER and any XER, we have D[fg, x; h, y, w] = D[ƒ, x; h, y, w]g(x) + f(x)D[g, x; h, y, w], where fg denotes the product of the functions f and g. Constant functions: For any h>0, any CER, any YER ¹+1, any WER n+1 and any XER, we have D[c, x; h, y, w] = 0. n+1 Linearity in the weights: For any h>0, any s,ter, any feC (R,R), any YER' D[f, x; h, y, sv + tw] = 1, any v,WER n+1 and any XER, we have sD[f, x; h, y, v] + tD[f, x; h, y, w].
Linearity in the function: For any h>0, s,ter, any f,g=C(R,R), any YER ¹+1, any wer n+1 and any XER, we have D[sf + tg, x; h, y, w] = sD[f, x; h, y, w] + tD[g, x; h, y, w]. Product rule: For any h>0, any f,geC∞ (R,R), any YER' n+1 n+1 any WER and any XER, we have D[fg, x; h, y, w] = D[ƒ, x; h, y, w]g(x) + f(x)D[g, x; h, y, w], where fg denotes the product of the functions f and g. Constant functions: For any h>0, any CER, any YER ¹+1, any WER n+1 and any XER, we have D[c, x; h, y, w] = 0. n+1 Linearity in the weights: For any h>0, any s,ter, any feC (R,R), any YER' D[f, x; h, y, sv + tw] = 1, any v,WER n+1 and any XER, we have sD[f, x; h, y, v] + tD[f, x; h, y, w].
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
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Which of the following statements are correct?
![n+1
', any WER and any XER, we have
a. Linearity in the function: For any h>0, s,ter, any f,geC (R,R), any YER +1,
D[sf + tg, x; h, y, w] = sD[f, x; h, y, w] + tD[g, x; h, y, w].
b. Product rule: For any h>0, any f,geC (R,R), any YER "+1, any WER n+1 and any XER, we have
D[fg, x; h, y, w] = D[f, x; h, y, w]g(x) + f(x)D[g, x; h, y, w],
where fg denotes the product of the functions f and g.
O c. Constant functions: For any h>0, any CER, any YER
n+1
n+1
', any WER
and any XER, we have
D[c, x; h, y, w] = 0.
O d. Linearity in the weights: For any h>0, any s,ter, any fεC∞(R,R), any YER 1, any v,WER n+1 and any XER, we have
n+1
D[f, x; h, y, sv + tw] = sD[ƒ, x; h, y, v] + tD[f, x; h, y, w].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe587adf5-17b3-4d31-bfe0-bfcde8e60070%2Fdbbb1461-67d3-4b2c-ae6c-f08d65d140b0%2F5y2ie7l_processed.png&w=3840&q=75)
Transcribed Image Text:n+1
', any WER and any XER, we have
a. Linearity in the function: For any h>0, s,ter, any f,geC (R,R), any YER +1,
D[sf + tg, x; h, y, w] = sD[f, x; h, y, w] + tD[g, x; h, y, w].
b. Product rule: For any h>0, any f,geC (R,R), any YER "+1, any WER n+1 and any XER, we have
D[fg, x; h, y, w] = D[f, x; h, y, w]g(x) + f(x)D[g, x; h, y, w],
where fg denotes the product of the functions f and g.
O c. Constant functions: For any h>0, any CER, any YER
n+1
n+1
', any WER
and any XER, we have
D[c, x; h, y, w] = 0.
O d. Linearity in the weights: For any h>0, any s,ter, any fεC∞(R,R), any YER 1, any v,WER n+1 and any XER, we have
n+1
D[f, x; h, y, sv + tw] = sD[ƒ, x; h, y, v] + tD[f, x; h, y, w].
![Consider the general numerical differentiation formula
with h>0 and certain constants
D[f, x; h, y, w]
'=
and
Σwk f(x + ykh)
k=0
y = (yo, ..., Yn) € R¹+¹
w := (wo, ..., w₁) € R¹+¹.
The differentiation formulas in Example 6.2 are particular examples of this general formula. We wish to approximate
f'(x) ≈ D[f, x; h, y, w].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe587adf5-17b3-4d31-bfe0-bfcde8e60070%2Fdbbb1461-67d3-4b2c-ae6c-f08d65d140b0%2Fafc8l7c_processed.png&w=3840&q=75)
Transcribed Image Text:Consider the general numerical differentiation formula
with h>0 and certain constants
D[f, x; h, y, w]
'=
and
Σwk f(x + ykh)
k=0
y = (yo, ..., Yn) € R¹+¹
w := (wo, ..., w₁) € R¹+¹.
The differentiation formulas in Example 6.2 are particular examples of this general formula. We wish to approximate
f'(x) ≈ D[f, x; h, y, w].
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