30 Jun A FUNCTION HAS AN X SQUARED TERM, AN X TERM, AND A CONSTANT
A function has an x squared term, an x term, and a constant
Question 1
At a value of x, the first derivative equals 0 and the second derivative is positive. At this point the function has reached
a local minimum
a local maximum
a global minimum
a global maximum
Question 2
A function has an x squared term, an x term, and a constant. The function has a single local/global maximum. The coefficient on the x squared term is
positive
zero
negative
unknown from this information
Question 3
A function has two critical values. It is possible that both of these provide relative minima.
True
False
Question 4
Let x be the number of units produced. The company wants to know how many units to make in order to maximize profit. Remember that Profit = Revenue Cost. At the optimal value of x, R(x) > C(x).
True
False
Question 5
When the derivative is equal to 0 at a value of x
the function will be equal to 0 at that value of x
the function has a critical point at that value of x
the second derivative will be 0 at that value of x
Each of the above is true.
Question 6
In this example, f(5) = 20 and f(6) = 25. In addition, f (5) > f (6). What can be said about f ” (5)?
It is positive
It is zero
It is negative
There is no way to tell from this information.
Question 7
The economic order quantity is the number of units to produce in order to minimize the sum of ordering costs and inventory holding costs.
True
False
Question 8
The integral of a marginal cost function is the total cost function itself (subject to the choice of the constant).
True
False
Question 9
The derivative of the anti-derivative of a function is the function itself.
True
False
Question 10
The antiderivative of the derivative of a function is the function itself.
True
False
Question 11
A function describes how the number of customers will grow over time. The only variable in the function is t, which stands for time in months. To find how many customers there will be after 10 months,
plug t = 10 into the function
plug t = 10 into the derivative of the function
find the definite integral of the function between t = 0 and t = 10
plug t = 10 into the second derivative of the function
Question 12
From BUS 205 you know that the area under any normal curve is 1.00. To prove this theoretically using calculus, you would evaluate
the equation of the standard normal curve from z = -3 to z = +3
the definite integral of the normal curve from x = negative infinity to x = positive infinity
the derivative of the normal curve at x = 0
the integral of the normal curve for positive values of x.
Question 13
To calculate the value of a definite integral, you must know the value of the constant term.
True
False
Question 14
A function describes the rate at which workers can produce an item. The variable x measures how many hours the worker has been working that day. If the worker begins the shift at 8:00, then the function at x = 1 would describe the production rate at 9:00. To find how many units are produced during the first four hours of the workers shift, find the definite integral of the function between x = 0 and x = 4.
True
False
Question 15
A company has developed a function that describes its profit over time. The x variable measures time. If the first derivative of the function is negative at a value of x, it means that the company is not making a profit at that point in time.
True
False
Section 2
1. The derivative of a function is given by
How does the original function behave at x = 2?
a. The original function is increasing at x = 2.
b. The original function is decreasing at x = 2.
c. The original function has reached a relative maximum at x = 2.
d. The original function is equal to 54 when x = 2.
2. The function below approximates the weekly box office receipts for a popular movie, where x = the number of weeks the movie has been playing.
What is the rate of change of weekly receipts per theater after 10 weeks?
a. Receipts are 5207.8
b. Receipts are growing at 5207.8 per week
c. Receipts are shrinking by 705.88 per week
d. Receipts are shrinking by 527 per week
Use this situation for questions 3 and 4.
The cost to produce x units of a product is given by the formula
3. What is the cost of producing 8 units?
a. 277
b. 312
c. 35
d. 51
4. When we are already producing 8 units, what do we estimate as the cost of the 9th unit?
a. 363
b. 34
c. 38
d. 51
5. At what value of x does this function reach a critical point, and what is it?
a. x = 2.5, maximum
b. x = 2.5, minimum
c. x = 2.5, point of inflection
d. there is no real value of x that causes the function to reach a critical point
6. The selling price of a product is $400, and the manufacturer is able to sell every unit it makes. The cost of producing x units is given by this formula:
How many units should be produced in order to maximize profit?
a. x = 25
b. x = 4
c. x = 100
d. x = 1000
7. Find the derivative of this function
a.
b.
c.
d.
8. Find the derivative of this function.
f(x)=3x^0.5-4x^(-2)
a. 3x^(-1.5)+8x^(-3)
b. 1.5x^(-0.5)+8x^(-3)
c. 1.5x^0.5+8x^(-1)
d. 1.5x^1.5+8x^(-1)
9. Find the derivative of this function
a.
b.
c.
d.
10. Consider the revenue functionRevenue(t)=t^2-12t+27. How fast is revenue changing at year 5?
a. -2
b. 4
c. -8
d. 8
Use this information for questions 11 15.
A function has the characteristics listed in the bullets below. Use the characteristics to construct a rough sketch of the function using this x axis:
_________________________________________________________________________________
-20 -15 -10 -5 0 5 10 15 20
The function has three critical points
When x = -20, the first derivative is positive
When x = -10, the first derivative is zero and the second derivative is negative
When x = 5, the first derivative is zero
When x = 15, the first derivative is zero
Indicate whether each of these statements is true or false
11. T F When x = 0, the first derivative is positive
12. T F When x = 5 the second derivative is positive
13. T F When x = 15 the second derivative is positive
14. T F The basic shape of this curve is more like an M than a W
15. T F For any value of x larger than 15, the first derivative will always be negative
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