please write down all work I know it’s a lot of problems but they are relatively easy. I don’t understand them though so it would be great if you can show the work step by step.

Week 1 Activity 1: Section 1 (Introduction to Functions)

Math 111, Fall ’20, Henderson

Name:

Purpose and Instructions: Consult the “Week 1 Activity” assignment on Canvas for complete information about instructions, the purpose of this homework set, due date, grading rubric and acceptable

communication.

Routine Exercises: The following exercises are designed to give you an idea of the content and

difficulty level of the Canvas and/or written component of Exams. Occasionally Routine exercises will

tell you what the answer is in order to stress that the most important skill is showing a complete process.

√

8−t

.

1. Let h(t) =

t + 12

(a) Compute h(−17). [Hint: you should get −1 as your answer.]

(b) Write h(8 − c2 ) in its simplest form, when c is a positive constant. [Hint: you should get

c

as your answer.]

20 − c2

1

Week 1 Activity 1: Section 1 (Introduction to Functions)

Math 111, Fall ’20, Henderson

2. The percentage of registered voters that vote in an election is viewed as a function of the effort

expended to “get out the vote”. One measure of effort is money spent on marketing, and a model

finds that

60r

M (r) =

100 − r

million dollars are spent on marketing when r% of registered voters actually vote.

(a) What is the (mathematical) domain of this function?

(b) What is the practical domain of the function?

(c) Find and write a sentence interpreting the value of M (25) in the applied context. Include

units.

(d) According to this model, what percentage of registered voters actually vote when 60 million

dollars are spent on marketing?

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Week 1 Activity 1: Section 1 (Introduction to Functions)

Math 111, Fall ’20, Henderson

3. Below are provided graphs of three benchmark percentiles for the weights of infant (at most 24

months old) girls: 97th percentile, 50th percentile, and 3rd percentile. A girl at the nth percentile

weighs more than n percent of other baby girls. Let’s define functions W1 (m), W2 (m) and W3 (m)

to be the weight, in pounds, of an infant girl at age m months in the 97th percentile, the weight

of a girl at age m in the 50th percentile, and the weight of a girl at age m in the 3rd percentile,

respectively.

32 weight (pounds)

28

W1

24

20

W2

16

12

8

W3

4

age (months)

3

6

9

12 15 18 21 24

(a) Estimate and interpret the values of W1 (6), W2 (6), and W3 (6).

(b) In each case, estimate the value(s) of m so that the statement is true, or state that no such

value exists

(i) W1 (m) = 22

(ii) W2 (m) = 22

(iii) W3 (m) = 22

(c) Which is larger W1 (12) − W3 (12) or W1 (24) − W3 (24)? What does this imply about infants

as they grow older?

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Week 1 Activity 1: Section 1 (Introduction to Functions)

Math 111, Fall ’20, Henderson

4. Write a formula for the function described, including naming the function and its variables (written

in bold). Write the formula in terms of the single input variable.

The total mass of a bacteria population is proportional to the number of bacteria

in the population. When there are 107 bacteria, the total mass is 3000 picograms.

4

Week 1 Activity 1: Section 1 (Introduction to Functions)

Math 111, Fall ’20, Henderson

Challenge Exercises: The following exercises are of a greater difficulty than the earlier ones,

though still matched to our course objectives. These exercises are not intended to prepare you for

test questions, instead they expose you to more complex, real-world scenarios. You may struggle

more with these questions than the Routine exercises. Remember your problem solving strategies!

Read carefully and repeatedly. What words are familiar in the problem statement? What terms

have been defined in the class, versus what is being provided to you within the exercise itself?

Who can you work with for assistance?

5. The human heart beats based on electrical signals passed between two specific nodes within the

heart, the sinoatrial (SA) node and the atrioventricular (AV) node. For a particular person, a

model for the potential (that is, the voltage) between these nodes is

(

0.5v + 25 , if 0 ≤ v ≤ 60

P (v) =

,

0.5v

, if v > 60

where P (v) is the new voltage (in millivolts) after 1 second, given a previous voltage v. The

function P (v) is called an “updating function” for the voltage of the heart because it gives you an

“update” on what the new voltage is after each second.

(a) Calculate and interpret, including units, the value of P (80). Do the same for P (48).

(b) What input voltage is required in order to have the output be 75 mV? Is there more than

one answer?

(c) The system is said to be in equilibrium at any value of v which is a solution to the equation

P (v) = v. Does this system have an equilibrium, and if so, where?

5

Week 1 Activity 1: Section 1 (Introduction to Functions)

Math 111, Fall ’20, Henderson

(d) When a voltage lands on the first branch of the function, the heart bumps up the voltage, and

the heart beats. When the voltage is larger than the threshold voltage, in this case 60 mV,

the heart does not beat and the voltage is reduced. It turns out that having an equilibrium

(a term which was defined in part (c) above) implies a healthy heart: it means that the heart

will continue to beat regularly every time it receives a signal to do so.

i. Does this heart beat regularly?

ii. Begin to test your claim from part (i) above by picking a value for v, let’s call it v0

and computing P (v0 ). We’ll call that answer v1 . Then compute P (v1 ) and call it v2 .

Continue this process until you have computed v0 through v5 .

iii. Based on your computations in part (ii), is it feasible that these numbers are getting

closer to the equilibrium you found in part (c)?

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