Question 1

Suppose the managers of a dairy company have modeled weekly production costs as  dollars for u units of dairy products. Weekly shipping cost for u units is given by  dollars
Calculate the total cost to produce and ship 4000 units in 1 week. Round to the nearest cent.

Accepted Answer

The answer of Suppose the managers of a dairy company...

Question 2

Evaluate  where  given  ,  ,  ,  .

Accepted Answer

The answer of Evaluate  where  given  ,...

Question 3

For the given pair of functions, write the composite function and its derivative in terms of one input variable.

Accepted Answer

The composite function is f(g(x)) where f(x) = e^x and g(x) = ln(x). 
So, the composite function is f(g(x)) = e^(ln(x)) = x. 
The derivative of f(g(x)) is the derivative of x, which is 1. 
Therefore, the derivative is 1.

Question 4

Using the product rule write the rate-of-change function.

Accepted Answer

The answer of Using the product rule write the rate-of-change...

Question 5

For the given pair of functions, write the composite function and its derivative in terms of one input variable.

Accepted Answer

The answer of For the given pair of functions, write...

Question 6

Differentiate the given function.

Accepted Answer

The answer of Differentiate the given function. 
...

Question 7

Find the derivative of  .

Accepted Answer

The answer of Find the derivative of  .
...

Question 8

Using the product rule write the rate-of-change function.

Accepted Answer

The answer of Using the product rule write the rate-of-change...

Question 9

Differentiate the given function.

Accepted Answer

The answer of Differentiate the given function. 
...

Question 10

Suppose the percentage of children living with their grandparents between 1970 and 2000 can be modeled by the equation  percent, t years after 1970. How rapidly on average did the percentage of children living with their grandparents grow between 1974 and 1993?

Accepted Answer

The answer of Suppose the percentage of children living with...

Question 11

Find the derivative of  .

Accepted Answer

The answer of Find the derivative of  .
...

Question 12

Find the derivative of  .

Accepted Answer

The answer of Find the derivative of  .
...

Question 13

The GDP of a certain country is $617 billion and is increasing by $22 thousand per person. The population of that country is 76 million and is increasing by 0.9 million people per year. How quickly is the GDP increasing?

Accepted Answer

The GDP is increasing by $22 thousand per person. With the population increasing by 0.9 million people per year, the increase in GDP from population growth is $22,000 * 900,000 = $19.8 billion per year.

Question 14

For the given pair of functions, write the composite function and its derivative in terms of one input variable.

Accepted Answer

The answer of For the given pair of functions, write...

Question 15

Differentiate the given function.

Accepted Answer

The answer of Differentiate the given function. 
...

Question 16

When t million dollars is invested in technology for a manufacturing plant, the plant needs w(t) workers to maximize production. Labor costs are L(w) million dollars when w workers are employed. When $5 million is invested in technology, 2400 workers are needed to maximize production, and labor needs are increasing by 200 workers per million dollars. It costs $32 million to employ 2400 workers. At 2400 workers, labor costs are increasing by approximately $0.024 per worker. Evaluate w(t) when $5 million is invested in technology, and write a sentence interpreting the value.

Accepted Answer

From the given information, we have:
- At $5 million investment, the plant needs 2400 workers to maximize production.
- Labor needs are increasing by 200 workers per million dollars. 
- Labor costs are increasing by approximately $0.024 per worker at 2400 workers.
- It costs $32 million to employ 2400 workers.

Let's find the labor cost function, L(w), at 2400 workers:
32 = L(2400)
So, L(w) = 32 + 0.024(w-2400)
Now, we can find the number of workers needed at $5 million investment as follows:
5 = (2400 + 200t)/1000
t = 2.5
So, at $5 million investment, the number of workers needed to maximize production is:
w(5) = 2400 + 200(2.5) = 2200

Therefore, the best choice is B, which states that when $5 million is invested in technology for a manufacturing plant, the plant needs 2200 workers to maximize production.

Question 17

For each of the composite functions, identify an inside function and an outside function and write the derivative with respect to x of the composite function.

Accepted Answer

Derivative: (e^x)(3x^2)

B) Inside Function: 3x+1; Outside Function: ln(x)
Derivative: (1/x)(3)

C) Inside Function: x^2-1; Outside Function: sqrt(x)
Derivative: (1/2sqrt(x))(2x)

D) Inside Function: 2x-1; Outside Function: cos(x)
Derivative: -sin(2x-1)(2)

E) There is no inside and outside function, so the question does not make sense in this case.

Answer: A, B, C 
 A, B, and C are the only options with clearly identifiable inside and outside functions. D has identifiable inside and outside functions but requires knowledge of the derivative of cosine. E is not a valid option since there is no inside and outside function.

Question 18

Find the derivative of  .

Accepted Answer

The derivative of the given expression is simply 0 because it is a constant term. Therefore, the correct answer is the option that only changes the fourth character in the hexadecimal number, which is the least significant byte. Changing this byte by one will result in a small change in the overall value of the number, indicating that it is the correct answer. The other options change the more significant bytes, resulting in a significant change in the value of the number, indicating that they are wrong.

Question 19

The revenue from the sale of x units of a commodity is r(x) Canadian dollars, and u(r) U.S. dollars is the equivalent value of r Canadian dollars. On September 8, 2009, $1 Canadian was worth $1.0764 U.S., and the rate of change of the U.S. dollar value was $0.925 U.S. per Canadian dollar. On the same day, sales were 470 units, producing revenue of $10,000 Canadian, and revenue was increasing by $4.2 Canadian per unit. Evaluate  of the given expression on September 8, 2009, and write a sentence interpreting the value.

Accepted Answer

We need to find the derivative of u(r(x)) with respect to x, where r(x) is the revenue function in Canadian dollars. Using the chain rule, we get:
u'(r(x)) * r'(x)
The current exchange rate is u(1) = 1.0764, so u'(1) = 0.925. Also, r(470) = 10000, and r'(x) = 4.2. Therefore, the expression we need to evaluate is:
0.925 * 4.2 = 3.88
This means that for each unit sold, the revenue in U.S. dollars is increasing by $3.88. Since the exchange rate is fixed, this is equivalent to an increase of $4.2 in Canadian dollars per unit. Therefore, the answer is A.

Question 20

The population (in millions) of the United States between 1970 and 2010 can be modeled as  million people where x is the number of decades after 1970.
The percentage of people in the United States who live in the Midwest between 1970 and 2010 can be modeled as  percent
Where x is the number of decades since 1970.
Write an expression for the number of people who live in the Midwest x decades after 1970.

Accepted Answer

The expression for the number of people who live in the Midwest x decades after 1970 is (11eaa894_64b1_a092_a158_8f2c042e8038_TB6775_11 million people) * (11eaa894_64b1_c7a3_a158_ef6d90dac4a2_TB6775_11 percent). Simplifying this expression gives us approximately 11eaa894_64b1_c7a6_a158_9bef82b01387_TB6775_11 million people. Therefore, the best choice is C.

Quiz 3: Determining Change: Derivatives

Suppose the managers of a dairy company have modeled weekly production costs as dollars for u units of dairy products. Weekly shipping cost for u units is given by dollars Calculate the total cost to produce and ship 4000 units in 1 week. Round to the nearest cent.

Evaluate where given , , , .

For the given pair of functions, write the composite function and its derivative in terms of one input variable.

Using the product rule write the rate-of-change function.

For the given pair of functions, write the composite function and its derivative in terms of one input variable.

Differentiate the given function.

Find the derivative of .

Using the product rule write the rate-of-change function.

Differentiate the given function.

Suppose the percentage of children living with their grandparents between 1970 and 2000 can be modeled by the equation percent, t years after 1970. How rapidly on average did the percentage of children living with their grandparents grow between 1974 and 1993?

Find the derivative of .

Find the derivative of .

The GDP of a certain country is $617 billion and is increasing by $22 thousand per person. The population of that country is 76 million and is increasing by 0.9 million people per year. How quickly is the GDP increasing?

For the given pair of functions, write the composite function and its derivative in terms of one input variable.

Differentiate the given function.

For each of the composite functions, identify an inside function and an outside function and write the derivative with respect to x of the composite function.

Find the derivative of .