Question

The graph of f(x) = 2^x + 1 is shown below. Explain how to find the average rate of change between x = 1 and x = 3.

Answer 1

Can you tell me what f(x) is when x=1 and when x=3?

Answer 2

@Michele_Laino

Answer 3

it is simple, we have to apply this formula:
\[r = \frac{{f\left( 3 \right) - f\left( 1 \right)}}{{3 - 1}}\]

Answer 4

now, we have:
\(f(3)= 2^3+1=...\)

Answer 5

ok:)

Answer 6

please continue, I will guide you to final answer

Answer 7

\(f(3)=...?\)

Answer 8

1?

Answer 9

hint:
\[f\left( 3 \right) = {2^3} + 1 = 8 + 1 = ...?\]
since \(2^3=8\)

Answer 10

https://www.desmos.com/calculator/d7rsqv1ntm \
This is what the graph looks like and isn't it asking us to find the slope then??

Answer 11

yes! Better is that we have to find an average rate, because, for this function, the term "slope" alone has no meaning

Answer 12

ok:) So what do we do?

Answer 13

please complete:
\(f(3)=8+1=...?\)

Answer 14

9

Answer 15

correct!

Answer 16

now, we have to compute this:
\(f(1)=2^1+1=2+1=...\)

Answer 17

3

Answer 18

that's right!
so after a substitution, into my formula above, we get:
\[r = \frac{{f\left( 3 \right) - f\left( 1 \right)}}{{3 - 1}} = \frac{{9 - 3}}{{3 - 1}} = ...?\]

Answer 19

6/2=3

Answer 20

that's right!! :)

Answer 21

now what?

Answer 22

we have completed the requested solution

Answer 23

did we? Ok so the average rate of change between x=1 and x=3 is 3?

Answer 24

yes!

Answer 25

Thanks!

Answer 26

:)

Answer 27

Can I ask one more??

Answer 28

ok!

Answer 29

A biologist is comparing the growth of a population of flies per week to the number of flies a lizard will consume per week. She has devised an equation to solve for which day (x) the lizard would be able to eat the entire population. The equation is 3x = 5x − 1.

Explain to the biologist how she can solve this on a graph using a system of equations

Answer 30

I got 2,9 as the answer:)

Answer 31

I think that the requested system is like below:
\[\left\{ \begin{gathered}
y = 3x \hfill \\
y = 5x - 1 \hfill \\
\end{gathered} \right.\]
since we need to introduce a new variable \(y\)

Answer 32

yes:) but it's actually 3^x

Answer 33

sorry :)

Answer 34

y=3^x<---log function
y=5x-1

Answer 35

ok! then we have to solve this system:
\[\left\{ \begin{gathered}
y = {3^x} \hfill \\
y = 5x - 1 \hfill \\
\end{gathered} \right.\]

Answer 36

\[\huge \left\{ \begin{gathered}
y = {3^x} \hfill \\
y = 5x - 1 \hfill \\
\end{gathered} \right.\]

Answer 37

yes:)

Answer 38

https://www.desmos.com/calculator/mtmjle2eyr this is the graph

Answer 39

so, we have to draw both graphs of the functions \(\Large y=3^x\) and \(\Large y=5x-1\)

Answer 40

https://www.desmos.com/calculator/mtmjle2eyr Yes:)

Answer 41

correct! Your answer is right!

Answer 42

2,9? Yay!

Answer 43

yes! Since we are searching for a number of days \(x\), so \(x\) has to be a whole number

Answer 44

cool! I have this LAST question that I'm really confused on :(

Answer 45

ok!

Answer 46

John has taken out a loan for college. He started paying off the loan with a first payment of $100. Each month he pays, he wants to pay back 1.1 times as the amount he paid the month before. Explain to John how to represent his first 20 payments in sigma notation. Then explain how to find the sum of his first 20 payments, using complete sentences. Explain why this series is convergent or divergent.

Answer 47

those payments starts with $100, and at the second day the payment is \(1.1 \cdot 100\)
at the third day it is \(1.1 \cdot (1.1 \cdot 100))=1.1^2 \cdot 100\), at the fourth day such payment is \(1.1 (1.1^2 \cdot 100)=1.1^3 \cdot 100\)
am I right?

Answer 48

yes:)

Answer 49

sorry! I meant months not days

Answer 50

It's okay:)

Answer 51

now what?

Answer 52

so at the fourth month the total payment is:
\[\Large \begin{gathered}
100 + 1.1 \cdot 100 + {1.1^2} \cdot 100 + {1.1^3} \cdot 100 = \hfill \\
\hfill \\
= \sum\limits_{x = 1}^4 {{{1.1}^{x - 1}} \cdot 100} \hfill \\
\end{gathered} \]

Answer 53

arent we suppose to find his first 20 payments?

Answer 54

so then the 4 becomes 20 at the top?

Answer 55

If we consider the initial $100 as the first payment then yes, we have to replace 4 with 20

Answer 56

ok:)

Answer 57

so what would be the answer?

Answer 58

so, in that case total payment is:
\[\huge P = \sum\limits_{x = 1}^{20} {{{1.1}^{x - 1}} \cdot 100} \]

Answer 59

now, we have to see if such series is convergent

Answer 60

ok:)

Answer 61

I think it's convergent

Answer 62

if the upper limit is \(+ \infty\) and not \(20\), then such series is divergent

Answer 63

yeah, but the upper limit is 20. And now we have tto solve it

Answer 64

5727.23?

Answer 65

yes! In order to do that, we can rewrite such series as below:
\[\Large P = \sum\limits_{x = 1}^{20} {{{1.1}^{x - 1}} \cdot 100} = \frac{{100}}{{1.1}} \cdot \sum\limits_{x = 1}^{20} {{{1.1}^x}} \]

Answer 66

please wait, I'm checking your result...

Answer 67

ok:)

Answer 68

I got this:
\[\Large P = \sum\limits_{x = 1}^{20} {{{1.1}^{x - 1}} \cdot 100} = \frac{{100}}{{1.1}} \cdot \sum\limits_{x = 1}^{20} {{{1.1}^x}} \cong 5,727.5\]

Answer 69

yay! Thank-you so much<3

Answer 70

:)

Answer 71

i need help on 1 more:)

Answer 72

A field test for a new exam was given to randomly selected seniors. The exams were graded, and the sample mean and sample standard deviation were calculated. Based on the results, the exam creator claims that on the same exam, nine times out of ten, seniors will have an average score within 4% of 70%.

Is the confidence interval at 90%, 95%, or 99%? What is the margin of error? Calculate the confidence interval and explain what it means in terms of the situation.

Answer 73

i think it's 90%?

Answer 74

I'm very sorry, I don't know this answer :(

Answer 75

why do you think it is 90%

Answer 76

um well I guessed :) Can you google the answer?

Answer 77

I don't know. I think nine times out of ten, it is equivalent to 9/10, nevertheless I'm not sure, please ask to another helper

Answer 78

@dan815 please can you help here?

Answer 79

Thanks for trying:)

Answer 80

:)

Answer 81

yeah 90% confidence interval

Answer 82

bUT what would the answer be??