The data in the table below, also shown in the figure, is taken from the CDC growth tables for children and adolescents, at the 50th percentile, for ages t given in years and heights H(t) in cm (to one decimal place). Age Height 2 85.0 3 93.9 4 5 6 7 8 9 100.8 107.7 114.7 121.5 127.6 132.9 Age Height 10 138.0 11 144.0 12 151.2 13 157.2 14 160.4 15 161.9 16 162.5 17 162.9 H'(a) = lim At→0 Height H (cm) 160 150 (i) H'(10) D+, where D+ 140 130 120 110 100 CDC growth chart for girls (50th percentile) We know that the growth rate (rate of change of height) at some value t = a that is, the derivative H'(a)= - is given by the limit It=a H(a + At) - H(a) At In situations where we do not have a formula for the function H(t), but only data at some discrete values of t, we can estimate (approximate) the derivative by taking "small" values of At, subject to the available data points; in this problem we will consider some different ways to do this. (a) Use the above data to estimate the growth rate H'(10) of girls at age 10 (be sure to include appropriate units). Do this in the following three ways: H(10 + At) - H (10) At - this is called the forward difference formula. 8 10 Aget (years) H(10) — H(10 - At) At 12 (ii) H'(10) D, where D_ this is called the backward difference formula. (Observe that this is the same as calculating (iii) Now take the average of the above two estimates: (check) H' (10) Do, where Do where Do = (D+ + D_) this is called the centred difference formula. for At = 1 for At = 1 H(10+h)-H(10) for h = -1). H(10 + At) - H(10 - At) 2At
The data in the table below, also shown in the figure, is taken from the CDC growth tables for children and adolescents, at the 50th percentile, for ages t given in years and heights H(t) in cm (to one decimal place). Age Height 2 85.0 3 93.9 4 5 6 7 8 9 100.8 107.7 114.7 121.5 127.6 132.9 Age Height 10 138.0 11 144.0 12 151.2 13 157.2 14 160.4 15 161.9 16 162.5 17 162.9 H'(a) = lim At→0 Height H (cm) 160 150 (i) H'(10) D+, where D+ 140 130 120 110 100 CDC growth chart for girls (50th percentile) We know that the growth rate (rate of change of height) at some value t = a that is, the derivative H'(a)= - is given by the limit It=a H(a + At) - H(a) At In situations where we do not have a formula for the function H(t), but only data at some discrete values of t, we can estimate (approximate) the derivative by taking "small" values of At, subject to the available data points; in this problem we will consider some different ways to do this. (a) Use the above data to estimate the growth rate H'(10) of girls at age 10 (be sure to include appropriate units). Do this in the following three ways: H(10 + At) - H (10) At - this is called the forward difference formula. 8 10 Aget (years) H(10) — H(10 - At) At 12 (ii) H'(10) D, where D_ this is called the backward difference formula. (Observe that this is the same as calculating (iii) Now take the average of the above two estimates: (check) H' (10) Do, where Do where Do = (D+ + D_) this is called the centred difference formula. for At = 1 for At = 1 H(10+h)-H(10) for h = -1). H(10 + At) - H(10 - At) 2At
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter7: Integration
Section7.CR: Chapter 7 Review
Problem 88CR
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Transcribed Image Text:The data in the table below, also shown in the figure, is taken from the CDC growth tables for
children and adolescents, at the 50th percentile, for ages t given in years and heights H(t) in cm (to
one decimal place).
Age Height
2
85.0
93.9
3
4
5
6
7
8
9
100.8
107.7
114.7
121.5
127.6
132.9
Age Height
10
138.0
11
144.0
12
151.2
13
157.2
14 160.4
15 161.9
16 162.5
17
162.9
H'(a) = lim
At→0
Height H (cm)
160
150-
(i) H' (10) D+, where D+
140
(ii) H'(10) D_, where D_
130
120
110-
100-
CDC growth chart for girls (50th percentile)
We know that the growth rate (rate of change of height) at some value t = a - that is, the derivative
H'(a) = a - is given by the limit
H(a + At) - H(a)
At
H(10 + At) - H(10)
At
- this is called the forward difference formula.
10
Age (years)
In situations where we do not have a formula for the function H(t), but only data at some discrete values of
t, we can estimate (approximate) the derivative by taking "small" values of At, subject to the available data
points; in this problem we will consider some different ways to do this.
12
(a) Use the above data to estimate the growth rate H'(10) of girls at age 10 (be sure to include
appropriate units). Do this in the following three ways:
H(10) H(10- At)
At
(iii) Now take the average of the above two estimates:
(check)
H'(10) ≈ Do, where Do = (D+ + D_)
this is called the centred difference formula.
14
16
for At = 1
for At = 1
- this is called the backward difference formula.
(Observe that this is the same as calculating (10+h)-H(10) for h = -1).
H(10 + At) - H (10 - At)
2At
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