Textbook Question
Composite functions
Let ƒ(x) = x³, g (x) = sin x and h(x) = √x .
Evaluate h(g( π/2)).
Ch. 1 - Functions
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 1, Problem 19
Yeast growth Consider a colony of yeast cells that has the shape of a cylinder. As the number of yeast cells increases, the cross-sectional area A (in mm²) of the colony increases but the height of the colony remains constant. If the colony starts from a single cell, the number of yeast cells (in millions) is approximated by the linear function N(A) - CₛA, where the constant Cₛ is known as the cell-surface coefficient. Use the given information to determine the cell-surface coefficient for each of the following colonies of yeast cells, and find the number of yeast cells in the colony when the cross-sectional area A reaches 150 mm². (Source: Letters in Applied Microbiology, 594, 59, 2014)
The scientific name of baker’s or brewer’s yeast (used in making bread, wine, and beer) is Saccharomyces cerevisiae. When the cross-sectional area of a colony of this yeast reaches 100 mm², there are 571 million yeast cells.
Verified step by step guidance1
Identify the given information: When the cross-sectional area A = 100 \(\text{ mm}\)^2, the number of yeast cells N = 571 \(\text{ million}\).
Use the linear function N(A) = C_s A to express the relationship between the number of yeast cells and the cross-sectional area.
Substitute the given values into the equation: 571 = C_s \(\times\) 100.
Solve for the cell-surface coefficient C_s by dividing both sides of the equation by 100.
Use the determined value of C_s to find the number of yeast cells when A = 150 \(\text{ mm}\)^2 by substituting into the equation N(A) = C_s \(\times\) 150.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Linear Functions
A linear function is a mathematical expression that describes a relationship between two variables, typically in the form of y = mx + b, where m is the slope and b is the y-intercept. In the context of the yeast growth question, the number of yeast cells N(A) is expressed as a linear function of the cross-sectional area A, indicating that as A increases, N(A) changes at a constant rate determined by the cell-surface coefficient Cs.
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Linearization
Cell-Surface Coefficient (Cs)
The cell-surface coefficient (Cs) is a constant that quantifies the relationship between the cross-sectional area of a yeast colony and the number of yeast cells it contains. It represents how many millions of cells correspond to each unit of area (mm²). Understanding Cs is crucial for calculating the number of cells in a colony based on its area, as it directly influences the linear function N(A).
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09:07Example 1: Minimizing Surface Area
Cross-Sectional Area
The cross-sectional area of a cylindrical colony of yeast refers to the area of a slice taken perpendicular to the height of the cylinder. It is a critical factor in determining the growth of the yeast colony, as the problem states that while the area increases, the height remains constant. This area is essential for applying the linear function N(A) to find the number of yeast cells at specific cross-sectional measurements.
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09:07Example 1: Minimizing Surface Area
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