H(x) = 3 + 4* (a) What is the net change in the height of the plant over the first 2 months after the experiment begins? Include proper units, (b) What is the average rate of change of the height of the plant over the first 2 months after the experiment begins? Include proper units. (c) Estimate the instantaneous rate of change of the height of the plant 2 months after the experiment begins. Include proper units.
H(x) = 3 + 4* (a) What is the net change in the height of the plant over the first 2 months after the experiment begins? Include proper units, (b) What is the average rate of change of the height of the plant over the first 2 months after the experiment begins? Include proper units. (c) Estimate the instantaneous rate of change of the height of the plant 2 months after the experiment begins. Include proper units.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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The height ( in inches) of a plant x months after an experiment begins is given by
![### Growth Function Analysis of Plant Height
Consider the function representing the height of a plant over time:
\[ H(t) = 3 + 4^t \]
Where \( H(t) \) is the height of the plant in centimeters (cm) and \( t \) is the time in months.
#### Questions:
**(a) What is the net change in the height of the plant over the first 2 months after the experiment begins? Include proper units.**
**(b) What is the average rate of change of the height of the plant over the first 2 months after the experiment begins? Include proper units.**
**(c) Estimate the instantaneous rate of change of the height of the plant 2 months after the experiment begins. Include proper units.**
#### Analysis:
1. **Net Change:**
To find the net change in height over the first 2 months, calculate the height at \( t = 2 \) and subtract the initial height at \( t = 0 \):
\[
\text{Net Change} = H(2) - H(0)
\]
\[
H(0) = 3 + 4^0 = 3 + 1 = 4 \, \text{cm}
\]
\[
H(2) = 3 + 4^2 = 3 + 16 = 19 \, \text{cm}
\]
\[
\text{Net Change} = 19 \, \text{cm} - 4 \, \text{cm} = 15 \, \text{cm}
\]
2. **Average Rate of Change:**
Calculate the average rate of change over the first 2 months as follows:
\[
\text{Average Rate of Change} = \frac{H(2) - H(0)}{2 - 0}
\]
\[
= \frac{19 \, \text{cm} - 4 \, \text{cm}}{2 \, \text{months}} = \frac{15 \, \text{cm}}{2 \, \text{months}} = 7.5 \, \text{cm/month}
\]
3. **Instantaneous Rate of Change:**
Estimate the instantaneous rate of change at \( t = 2 \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F48bba008-5eee-4157-b489-6f90113c3fbd%2Ff33262c6-b27e-4f8a-a9a9-89bf43464204%2Fugg8ji_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Growth Function Analysis of Plant Height
Consider the function representing the height of a plant over time:
\[ H(t) = 3 + 4^t \]
Where \( H(t) \) is the height of the plant in centimeters (cm) and \( t \) is the time in months.
#### Questions:
**(a) What is the net change in the height of the plant over the first 2 months after the experiment begins? Include proper units.**
**(b) What is the average rate of change of the height of the plant over the first 2 months after the experiment begins? Include proper units.**
**(c) Estimate the instantaneous rate of change of the height of the plant 2 months after the experiment begins. Include proper units.**
#### Analysis:
1. **Net Change:**
To find the net change in height over the first 2 months, calculate the height at \( t = 2 \) and subtract the initial height at \( t = 0 \):
\[
\text{Net Change} = H(2) - H(0)
\]
\[
H(0) = 3 + 4^0 = 3 + 1 = 4 \, \text{cm}
\]
\[
H(2) = 3 + 4^2 = 3 + 16 = 19 \, \text{cm}
\]
\[
\text{Net Change} = 19 \, \text{cm} - 4 \, \text{cm} = 15 \, \text{cm}
\]
2. **Average Rate of Change:**
Calculate the average rate of change over the first 2 months as follows:
\[
\text{Average Rate of Change} = \frac{H(2) - H(0)}{2 - 0}
\]
\[
= \frac{19 \, \text{cm} - 4 \, \text{cm}}{2 \, \text{months}} = \frac{15 \, \text{cm}}{2 \, \text{months}} = 7.5 \, \text{cm/month}
\]
3. **Instantaneous Rate of Change:**
Estimate the instantaneous rate of change at \( t = 2 \)
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