The philosopher Alfred North Whitehead (1861-1947) postulated:

“It is a profoundly erroneous truism that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them.”

These drill questions are intended to help you learn to perform certain mathematical or modeling operations without thinking about them. Once you have accomplished this, you will be able to focus cognitive effort on those aspects of the modeling problem at hand that require knowledge of the setting, creativity, and higher-order skills in modeling.

Drill questions, by and large, are not about applications of modeling or higher-order skills. They are usually devoid of real-world context. Ideally, they are easy to answer once you have learned the topic. They are not “make-work” problems; think of them as weight-training exercises that strengthen a particular muscle group.

The drill questions are self-correcting in order to help you track your progress in developing skill. Don’t be tempted to blindly try answers until you get a positive response. That’s a complete waste of time, the ultimate in “make-work.”

Keep answering them until the answers come easily to you. On a second pass through the questions, clear the document memory as described in the “Remember to hand in your work …” box. That way you won’t have your previous answers to guide you.

Drill Questions: Functions

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Consider the following plot:

  1. What is the function output when the input is F? d05-1-a

    -15       -4.9       0       0.3       1.1      

  2. When the function output is -15, which letter corresponds to an appropriate input value? d05-1-b

    A       B       C       D       F      

Consider the following plot:

  1. What is the function output when the input is C? d05-2-a

    5.4       6.9       14      

  2. When the function output is 14, which letter corresponds to an appropriate input value? d05-2-b

    A       B       C      

With respect to the following plot …

  1. What is the function output when the input is B? d05-3-a

    -24.6       -20       -2.7       4.7       6.7      

  2. When the function output is -20, which letter corresponds to an appropriate input value? d05-3-b

    A       B       C       D       E      

Consider the following plot:

  1. What is the function output when the input is A? d05-4-a

    -18.3       -16       -13.8       -9.9      

  2. When the function output is -16, which letter corresponds to an appropriate input value? d05-4-b

    A       B       C       D      

Consider the following plot:

  1. What is the function output when the input is A? d05-5-a

    -14.6       -9       2      

  2. When the function output is -9, which letter corresponds to an appropriate input value? d05-5-b

    A       B       C      

Consider the following plot:

  1. What is the function output when the input is D? d05-6-a

    -0.8       -0.5       7       11.2      

  2. When the function output is 7, which letter corresponds to an appropriate input value? d05-6-b

    A       B       C       D      

Consider the following plot:

  1. What is the function output when the input is A? d05-7-a

    7.2       10       18.7      

  2. When the function output is 10, which letter corresponds to an appropriate input value? d05-7-b

    A       B       C      

Consider the following plot:

  1. What is the function output when the input is D? d05-8-a

    -1       1       3       5.6      

  2. When the function output is 1, which letter corresponds to an appropriate input value? d05-8-b

    A       B       C       D      

Consider the following plot:

  1. What is the function output when the input is C? d05-9-a

    -6.4       -5.9       -5       -3.3      

  2. When the function output is -5, which letter corresponds to an appropriate input value? d05-9-b

    A       B       C       D      

Rates of change, introduced in Section 3.5, are an important way of summarizing the relationships between quantities. Numerically, a rate of change is a quantity that relates two quantities of interest. Let’s call them A and B. The rate of change of A with respect to B is the ratio of the change in B divided into the corresponding change in A.

Rates of change are an important way of summarizing a model used in many circumstances. For example, the economists’ concepts of “marginal cost” or “elasticity” are rates of change.

Graphically, the rate of change of a function with respect to an input is the slope of the graph.

Each of the graphs in this section displays a function along with a set of red annotations that give the quantitative value of several different slopes.

Figure 1: A function annotated with slope lines

There are six slopes in the red annotation. You can measure the rate of change of the function with respect to the input by comparing the function to the slope annotations. For instance, near y=0 the function’s slope is about -1. The slope 0 is not shown in the annotation. It corresponds to a horizontal line.

Notice that the slope changes along with y. This is typical of functions used for modeling. So we can speak of the slope of a function at a particular input. Sometimes this “at a particular input” is called the “instantaneous” slope.” But we’ll get to that later in the course.

  1. What is the most positive rate of change for the function?
d05-Aa
-4       -2       0       2       4      

  1. At which of these inputs does that most positive rate of change occur?
d05-Ab
-2       -1       0       1      

  1. For what input is the rate of change practically zero?
d05-Ac
-2       0       1       2       3      

  1. What is the rate of change when y = -2?
d05-Ba
-6       -4       -2       2       4      

  1. What is the rate of change when y = 3?
d05-Bb
-6       -2       0       2       6      

  1. For what input is the rate of change practically zero?
d05-Bc
-3       0       1.5       2.5       3      

  1. What is the rate of change when y = 0?
d05-Ca
-4       -6       4       6       greater than 6      

  1. At what value of the input is the function output the greatest?
d05-Cb
-1.5       -0.5       0.5       1.5      

  1. What is the rate of change of the function when the input is at the value in question (b)?
d05-Cc
-4       -2       0       2       4      

  1. At what value of the input is the function output the smallest?
d05-Cd
-1.5       -0.5       0.5       1.5      

  1. What is the rate of change of the function when the input is at the value in question (b)?
d05-Ce
-4       -2       0       2       4      

  1. Reading the graph from left to right, at what value of the input does the function’s slope change from positive to negative?
d05-Da
1.75       2.75       5.5       7.5      

  1. Still reading from left to right, at what value of the input does the function’s slope change from negative to positive?
d05-Db
1.5       2.5       5.5       7.5      

  1. For which input is the slope of the function close to -1.5?
d05-Dc
1       2       3       4       5      

  1. Is there any input for which the function’s rate of change is -1? (Think before answering.)
d05-Ea
no       yes      

  1. At what input does the rate of change of the function go from negative to positive?
d05-Eb
-2.4       -1       -0.5       1.25      

  1. For what input is the rate of change of the function about -1?
d05-Fa
-4.5       -3       0       3      

  1. For what input is the rate of change of the function about +1?
d05-Fb
-2       -1       0       0.5       2      

How is the output value represented on a contour plot?

d05-output-rep

As the position along the horizontal axis.

As the position along the vertical axis.

By the labels along the contour lines.

The output isn’t shown.

For the function plotted just above …

  1. What is the output spacing between adjacent contours?
cp-space-3kws
1       2       3       4       5      

  1. What is the output spacing between adjacent labelled contours?
cp-labels-ekx
1       3       4       5       6      

  1. What is the minimum possible value for the function output on the input space shown. (Pick the closest answer.)
cp-min-3kws
-3       -1       0       2       3      

  1. What is the maximum possible value for the function output in the input space shown. (Pick the closest answer.)
cp-max-34s
12       13.5       15       17.5      

  1. Consider an input close to x = -3, z = 0, on the left side of the input space. Which of these is the best statement about the possible output for that input?
cp-sure-382

The output is less than 2.

The output is greater than 2.

The output is greater than 2 and less than 3.

The output is greater than 2 and less than 4.

  1. Is the landscape near point A a hilltop or the bottom of a bowl?
d05-A-dkwe
hilltop       bowl      

  1. Which point is at the lowest output value of the function shown?
d05-2-low
A       B       C       D       E      

  1. Which point is at the highest output value of the function shown?
d05-3-high
A       B       C       D       E      

  1. If you went on a walk on a straight path between points D and E, which of the following is an output value you would never see?
d05-4-high
-10.3       -9.8       -8.6       -8.1      

  1. If the output is the elevation of the landscape, walking from point C to E which of these is true?
d05-5-ew22

You are walking uphill.

You are walking on the level.

You are walking downhill.

Neither of the above.

  1. Walking from point F in a southwest (↙) direction, which is the best description for how your elevation is changing.
d05-6-ewu2

You are walking uphill.

You are walking on the level.

You are walking downhill.

You are going up and down.

Suppose you have set the input values so that the corresponding point lies exactly on some contour. Now you change the input values in such a way so that they always correspond to other points on the same contour. Let’s call this “moving along the contour.”

How does the output value change as the input moves along a contour?

d05-move

The output is unchanged.

The output always goes up.

The output always goes down.

The output goes up or down depending on whether the movement is clockwise or counter-clockwise.

Drill 5. 1 Consider a function Profit(price, weather) used by an ice cream shop owner to predict daily earnings.

  • Input 1: price (the cost of a cone, set by the owner).

  • Input 2: weather (the daily high temperature).

  • Internal Calculation: The model internally calculates demand (how many people buy ice cream) based on the price and weather. It then calculates profit by subtracting costs from revenue.

In a system diagram of this specific model, how would we classify the variable demand?

sts-1-k3s
Exogenous       Endogenous       An Argument       An Algorithm      

Drill 5. 2 You are examining a function Projectile_Distance(speed, angle) which calculates how far a ball travels based on the angle it is thrown and the speed at release.

If you slice this function at a fixed release speed, you will have created a function of just one input: angle. A graph of the function can be read to find the input angle required to hit a target exactly 50 meters away.

How many possible answers for angle should you expect to find?

dgf-10eks

Exactly 1

Exactly 2

Exactly 0

It might be any of the above

Drill 5. 3 Consider a function Total_Cost(Units, Unit_Price). You create two slices of this function to understand the behavior:

  • Slice A: You hold Unit_Price constant at $10 and vary Units from 1 to 100.
  • Slice B: You hold Units constant at 100 and vary Unit_Price from $10 to $1.

What is the geometric shape of the slice plots in Slice A and Slice B?

ats-1-clws

Slice A is curved; Slice B is curved.

Slice A is linear; Slice B is linear.

Slice A is linear; Slice B is horizontal.

Slice A is vertical; Slice B is linear.

Drill 5. 4 

graph RL
  A(x)
  B(y)
  C[w]
  D(z)
  A --> C
  B --> C
  D --> C
  
classDef exogenous fill:#fff,stroke-width:1px

class A exogenous
class B exogenous
class D exogenous

In the above diagram …

  1. Which quantities are endogenous?
dddd
w       x       y       z       None of them.      

  1. How many functions are indicated by the diagram?
dddd-2
0       1       2       3       4      

  1. Which quantity is the output of a function?
dddd-3
w       x       y       z       None of them.      

Drill 5. 5 According to Chapter 5, what is the definition of a mathematical function?

tlw-1-dkw

A mathematical concept that converts units associated with quantities.

A mathematical concept that connects an unknown variable to a desired solution.

A mechanism that transforms one or more inputs into a corresponding output.

A mechanism that transforms one or more inputs into a collection of outputs.

A graph that passes the horizontal line test.

A graph that has a horizontal and vertical axis.

Drill 5. 6 In the text’s example about sun position, cloudiness, and production, select the correct description:

srb-1-njvc

All three variables are endogenous.

The variables sun position and cloudiness are endogenous; production is exogenous.

The variables sun position and cloudiness are exogenous; production is endogenous.

All three variables are exogenous.

Drill 5. 7 Functions with multiple inputs can be analyzed using a technique called “slicing.” Which of the following statements accurately describes this process? Select any correct answer.

elr-1-jse

It involves holding one or more input variables constant at a fixed value.

It requires setting the function’s output to zero to solve for the intercepts.

It allows you to visualize the relationship between the output and a single input on a standard input vs output graph.

It calculates the average rate of change for all inputs simultaneously.

It enables some analysis of a multi-variable function as a single-variable function.

It combines all input variables into a single sum to determine the total magnitude.

Drill 5. 8 The words used to identify the various inputs to a function are called ____.

apb-1-hes

abstractions

algorithms

analyses

arguments

asymptotes

axes

Drill 5. 9  

  1. Recall the monthly_payment() function. xf(“fig-payment-r-slice”)is a slice plot that holdsprincipal` constant. Which of the following correctly describes the plot?
sbj-1-keiw

interest_rate as input, years as output.

years as input, interest_rate as output.

years as input, monthly_payment as output.

monthly_payment as input, years as output.

interest_rate as input, monthly_payment as output.

monthly_payment as input, interest_rate as output.

  1. Figure 5.5 is a similar slice plot. that holds principal constant. Which of the following correctly describes that plot?
sbj-2-keiw

interest_rate as input, years as output.

years as input, interest_rate as output.

years as input, monthly_payment as output.

monthly_payment as input, years as output.

interest_rate as input, monthly_payment as output.

monthly_payment as input, interest_rate as output.