Questions: Multiple Choice 1 point Suppose total egg production of a colony of shrimp is given by E(m)=s(m) · g(m), where s(m) is the colony size (in shrimp) and g(m) the laying capacity (in eggs per shrimp), at year m. What are the units of dE/dm ? shrimp per colony eggs per shrimp per year eggs per shrimp per colony eggs per year shrimp per colony per year shrimp per year Clear my selection Multiple Choice 2 points Product Rule is a method for taking the derivative of which type of function? A product of two functions. A quotient of two functions. A trigonometric function. A composition of two functions. A hyperbolic function.

Multiple Choice
1 point
Suppose total egg production of a colony of shrimp is given by E(m)=s(m) · g(m), where s(m) is the colony size (in shrimp) and g(m) the laying capacity (in eggs per shrimp), at year m. What are the units of dE/dm ?
shrimp per colony
eggs per shrimp per year
eggs per shrimp per colony
eggs per year
shrimp per colony per year
shrimp per year
Clear my selection

Multiple Choice
2 points

Product Rule is a method for taking the derivative of which type of function?
A product of two functions.
A quotient of two functions.
A trigonometric function.
A composition of two functions.
A hyperbolic function.

Transcript text: 14 Multiple Choice 1 point Suppose total egg production of a colony of shrimp is given by $E(m)=s(m) \cdot g(m)$, where $s(m)$ is the colony size (in shrimp) and $g(m)$ the laying capacity (in eggs per shrimp), at year $m$. What are the units of $\frac{d E}{d m}$ ? shrimp per colony eggs per shrimp per year eggs per shrimp per colony eggs per year shrimp per colony per year shrimp per year Clear my selection 15 Multiple Choice 2 points Product Rule is a method for taking the derivative of which type of function? A product of two functions. A quotient of two functions. A trigonometric function. A composition of two functions. A hyperbolic function.

Solution

Solution Steps

Solution Approach

To determine the units of $\frac{dE}{dm}$, we need to understand the units of $E(m) = s(m) \cdot g(m)$. Since $s(m)$ is the colony size in shrimp and $g(m)$ is the laying capacity in eggs per shrimp, the units of $E(m)$ are eggs. Therefore, the units of $\frac{dE}{dm}$ are eggs per year, as it represents the rate of change of egg production with respect to time.
The Product Rule is used for taking the derivative of a product of two functions. It states that if you have two functions $u(m)$ and $v(m)$, the derivative of their product is given by $(uv)' = u'v + uv'$.

Step 1: Determine the Units of $\frac{dE}{dm}$

The total egg production of a colony of shrimp is given by $E(m) = s(m) \cdot g(m)$, where $s(m)$ represents the colony size in shrimp and $g(m)$ denotes the laying capacity in eggs per shrimp. Therefore, the units of $E(m)$ are eggs. To find the units of $\frac{dE}{dm}$, we recognize that it represents the rate of change of egg production with respect to time. Thus, the units of $\frac{dE}{dm}$ are:

\[ \frac{dE}{dm} \text{ has units of } \text{eggs per year.} \]

Step 2: Identify the Application of the Product Rule

The Product Rule is a method used for taking the derivative of a product of two functions. Specifically, if $u(m)$ and $v(m)$ are two functions, the Product Rule states that:

\[ (uv)' = u'v + uv'. \]

This indicates that the Product Rule applies to the derivative of a product of two functions.