Deck 21: Calculus-Based Optimization

Question 1

To determine the slope of a nonlinear function at a point, we must find the slope of a line tangent to the point.

Answer 1

True

Question 2

The derivative of 2x² + 2x + 7 is

A)2x + 2.
B)1x + 7.
C)4x + 2.
D)x² + 7.

Answer 2

4x + 2.

Question 3

The derivative of 19x⁴ + 2x³ + 5x² + 1x + 9 is

A)19x³ + 2x² + 5x + 1.
B)4x³ + 3x² + 2x + 1.
C)12x² + 6x + 9.
D)76x³ + 6x² + 10x + 1.

Answer 3

76x³ + 6x² + 10x + 1.

Question 4

A zero value for the derivative of a nonlinear function always indicates the location of a maximum or minimum.

Question 5

The EOQ model is obtained by taking the second derivative of the revenue function.

Question 6

If, for a nonlinear function, the first derivative is equal to zero and the second derivative is equal to zero, we have both a maximum and a minimum occurring simultaneously.

Question 7

The slope of a nonlinear function varies depending on the point along the line at which it is measured.

Question 8

The points of inflection of the curve + + x + 12 are X = 1 and X = -1.

Question 9

The derivative of a constant is one.

Question 10

The slope of the line Y = 3 + 2x + 1 at the point (2,17)is 12.

Question 11

We can find a point of inflection by simply setting the second derivative of the nonlinear function equal to zero and solving.

Question 12

Total revenue is governed by the equation TR = 3 = + 7.Maximum revenue occurs when X = .

Question 13

The slope of the line Y = 4 is 1.

Question 14

The derivative of 5x³ + 3x² + 7x + 9 is

A)15x² + 6x + 7.
B)6x² + 7x + 9.
C)5x² + 3x + 7.
D)3x² + 2x + 7.

Question 15

The first step in attempting to find a minimum or maximum of a nonlinear function is to take the derivative of the function, set it equal to zero, and solve for x.

Question 16

The second derivative of the function 4 + 3 + 2 + x + 7 is 48 + 18x.

Question 17

At a minimum of a nonlinear function, the first derivative is equal to zero, and the second derivative is positive.

Question 18

The slope of a straight line is constant when measured at any point along the line.

Question 19

The second derivative tells about the slope of the first derivative.

Question 20

To maximize total revenues, one can set the derivative of the total revenue function equal to zero and solve.

Question 21

The second derivative of 7x⁴ + 6x³ + 5x + 3 is

A)7x³ + 6x² + 5.
B)28x³ + 18x² + 5x + 3.
C)28x³ + 18x² + 5.
D)84x² + 36x.

Question 22

The slope of the line Y = 3 + + 17 when X = 3 is

A)16.
B)17.78.
C)44.67.
D)46.67.

Question 23

The Dow Jones index reached dizzy heights of 22,000 but sadly the market was now in free fall thanks to an ill-phrased tweet.Free fall is governed by the equation S = + t + where S₀ is the initial position, in this case 22,000, and V₀ is the initial velocity, in this case 0.Acceleration due to gravity is represented in the equation with the letter a, which has the value a = 32.2.The time in seconds is given by the letter t.How fast is the Dow Jones index dropping one minute after the tweet?

A)1932 points/second
B)32.2 points/second
C)57960 points/second
D)1760 points/second

Question 24

Where are the critical points for the function y = 8 - 16 - 8x -12? What is the maximum of this function?

Question 25

Find the slope of the line Y = 4X-3 at the point where X = 17.

Question 26

The critical point for the function y = 7x² - 7x + 4 lies at

A)x = -1.
B)x = 1.
C)x = -1/2.
D)x = 1/2.

Question 27

The slope of the line Y = 4X-3 is

A)4.
B)3.
C)-4.
D)-3.

Question 28

Profit for the Dew Drop Inn Country Diner is given by p = -0.1q₂ + 1.80q + 100, where p is profit and q is the number of customers.What number of customers maximizes profit?

A)100
B)900
C)300
D)9

Question 29

The critical points for the function y = 8 - 16 - 8x - 12 are at

A) .
B) .
C) .
D) .

Question 30

Through rigorous quantitative analysis, Geoff develops a trading algorithm that will permit him to gain a competitive advantage over his previous dartboard method of selecting stocks.What he has noticed is that a stock's value at the end of the month can be accurately forecast using the equation Price_END = + Price_START where Price_START is the stock price at the start of the month and x is the number of days until the next full moon.At what point in time will the stock be at its maximum and minimum? If SBUX is trading at $50 per share at the start of the month, what profit can Geoff realize if he buys 100 shares at its minimum price and sells all 100 shares at its maximum price.Assume Geoff pays a $10 fee for the buy order and another $10 for the sell order.

Question 31

The second derivative of 8 + 7 + 3x + 9 is

A)48 + 35
+3.
B)960 + 420
.
C)240 + 140
.
D)320 + 120
+ 3
+ 9x.

Question 32

Find the point at which the following function is a maximum: - 2x² + 3x + 6.

A)x = 3
B)x = 1
C)x = 0
D)x = -3

Question 33

What is the second derivative of 7 + 6 + 3 +9?

Question 34

The derivative of is

A)
B)-
C)
D)

Question 35

Find the point at which the following function is a minimum: y = - 2x² + 3x + 6.

A)x = 3
B)x = 1
C)x = 0
D)x = -3

Question 36

Find the critical point for the function y = X³ + 60.This point is a(n)

A)maximum.
B)minimum.
C)inflection point.
D)slope

Question 37

Find the point at which the following function has an inflection point: y = - 2x² + 3x + 6.

A)3
B)1
C)2
D)0

Question 38

The critical point for the function y = 3x² + 5x + 2 is at

A)x = -5/6.
B)x = 3/5.
C)x = -3/2.
D)There is no critical point.

Question 39

Find the slope of the line Y = 3 + + 17 when X = 10.