# Project 4: Tides & Functions

For this project, your task is to come up with an equation that models the height of the water in Newtown Creek as a function of time.

To do this, you will need to collect lots of data! Because of this, groups may share data. Think about what data you’ll need to collect and what a good plan for collecting it is.

The project is due by the end of the day on Tuesday, April 24th.

Here are the groups:

MR2-1

• Ben, Josh, Richard, Meer
• Sydney, Sonicka, Alyssa, Zoë
• Sobohee, Rachel, Jane
• Kai, Alema, Islam
• Shilpa, Paul, Gabe, Stephanie

MR2-3

• Alex, Ashley, Dewan, Kate
• Max, Xavier, Qi
• Jackie, Chris, Ilinca, Emily
• Danielle, Jeisleen, Zak
• Kenneth, Mary, Brianna
• Izabela, Kendell, Maya, Balpreet

You will be evaluated primarily on the accuracy of your equation and on your description of your process. You also must include the data collected either as a table or a scatter plot or both. An excellent project will include an equation for a function that comes within an hour and within a foot of each data point and an explanation of how you applied knowledge of trigonometric graphs and transformations to fit the function to the data.

# Critical vocabulary for midterm

Can you define all these terms? Doing so should be excellent practice for the exam, as well as a chance to review topics we’ve focused on over the past six weeks.

Rational function

Exponential function

Logarithmic function

Asymptote

Zeroes

y-intercept

Logarithm

Inverse function

Logarithmic scale

Exponential growth

Exponential decay

Exponential growth rate

Exponential decay rate

Continuously compounded interest

# Solutions to Midterm Exam Review

1. 3

2. -2

3. -4

4. 4

5. 2

6. 3

7. 0

8. 3

9. 2

10. 0

11. $\frac{-39}{4}$

12. -48

13. $4 \log 2-8=\log 16-8$

14. $x=\frac{11}{4}$

15. $x=\frac{18}{11}$

16. $x\le7$

17. x<-1

18. $x=\{\frac{-1}{3},1\}$

19. x=5

20. x=2

21. $x\ge 4$

22. x=(3,103)

23. x=(1.5,5)

25. x=8

26. $x=-3+4\sqrt{2}$

27. $x=\frac{\ln \frac{1}{2}}{2}$

28. $x=-\ln 4$

29. $x=\log_5 3$

30. $x=\frac{\log_2 7}{\log_2 7-1}$

31. $x=\{ 0,\ln 4 \}$

32. x=1

33. $\frac{\ln 3}{0.2}$ hours

34. $y=2^{x-2}-2$

35. $10000\sqrt{10}$

36. Logarithmic scales allow us to express a huge range of intensity using manageable numbers.

37. $\log_2 9$ is larger because $2^3=8$, which is less than 9, and $5^3=125$, which is greater than 30. So $\log_2 9$ must be more than 3 and $\log_5 30$ must be less than 3.

38.

39.  x-intercept at x=-996

40. a. Rational functions can have vertical asymptotes, horizontal asymptotes, slant asymptotes, and curvilinear asymptotes.

b. All exponential functions have horizontal asymptotes but no other asymptotes.

c. All logarithmic functions have vertical asymptotes but no other asymptotes.

41.

42.

43. Horizontal asymptote at y=2. Vertical asymptotes at x=2 and x=-3.

44.

45.

46. Horizontal asymptote at y=-4. Vertical asymptotes at x=2 and x=-2.

# Project #3: Chemistry Sandwich

While in chem lab, you find a sandwich behind some beakers. You’re very hungry, having missed lunch, and are thinking about eating this sandwich that you found, but you don’t know how long it’s been there. Using a handy Geiger counter, you find that the sandwich has 98% of its carbon-14 remaining. Would you eat the sandwich? Explain your answer algebraically.

# Solutions to logarithmic equations classwork

1. $x=-\log 4$

2. $x=\frac{\ln 200-1}{2}$

3. $x=-\ln 4$

4. $x=\ln 3$

5. $x=\frac{1}{100}$

6. $x=-e^2+3$

7. $x=5$

8. $x=\frac{-1+\sqrt{9+4e}}{2}$

9. $x=43046721$

10. $x=\{101,1.1\}$

11. $x=64$

12. $x=\log_2 3$

# Classwork: EARTHQUAKE!

What was the difference in magnitude between the March 11th Tohoku earthquake and the August 23rd Virginia earthquake? How many times more intense was the Tohoku earthquake? How does this relate to logarithms?

# Classwork: Logarithmic Equations

Solve each equation.

1. $10^{-x}=4$

2. $e^{2x+1}=200$

3. $\frac{10}{1+e^{-x}}=2$

4. $e^{2x}-e^x-6=0$

5. $\log x=-2$

6. $2-\ln(3-x)=0$

7. $\log_2 3+\log_2 x=\log_2 5+\log_2 (x-2)$

8. $\ln(x-1)+\ln(x+2)=1$

9. $\log_2 (\log_3 x)=4$

10. $(x-1)^{\log(x-1)}=100(x-1)$

11. $\log_2 x+\log_4 x+\log_8 x=11$

12. $4^x-2^{x+1}=3$