Week 13 — Doubling, Halving, and Interest on Interest: A Short Exponential Report — Traditional alternative (no AI)

Instructors: publish this instead of the adaptive version if you prefer, and unpublish or delete the adaptive one.

Assignment 12 of the term — evaluate an exponential at a negative input, track a halving sample, price a phone that loses value, compound a savings balance, and tell someone new to money math why interest on interest matters.

This week you learned to tell a constant ratio from a constant difference, to read the initial value and the growth factor of an exponential function, to evaluate it with the exponent first, to tell growth from decay by the base, to turn a percent rate into a factor, and to compute compound interest. In Week 1 you learned that a negative exponent means a reciprocal, and in Week 3 you learned to evaluate a function and read its range; this week both return as exponential functions. In this assignment you work four short problems, interpret each one, and explain to someone new to money math why interest on interest matters. This week's lecture tutorial, quiz, discussion, and Model/Build Studio practise the same ideas from other angles.

The task

You will use four pieces of this week's work: the exponential function f(x) = 3 · 2ˣ, the 100-gram sample A(t) = 100 · (1 ⁄ 2)ᵗ that halves every period, the 1,200-dollar phone V(t) = 1200 · (0.75)ᵗ that loses 25% of its value each year, and 1,000 dollars at 6% interest compounded annually, A = 1000 · (1.06)ᵗ.

Work these four problems, showing your arithmetic:

1. Evaluate f(−2) for the exponential function f(x) = 3 · 2ˣ. What is its value?
2. A 100-gram sample halves every period, so A(t) = 100 · (1 ⁄ 2)ᵗ with t in half-lives. How many grams remain after 3 half-lives?
3. A 1,200-dollar phone loses 25% of its value each year, so V(t) = 1200 · (0.75)ᵗ. How many dollars is it worth after 2 years?
4. 1,000 dollars earns 6% interest compounded annually, so A = 1000 · (1.06)ᵗ. How many dollars are in the account after 2 years?

Beside each answer, add one brief interpretation: what that value shows and one use or limit of it. For problem 1, explain why a negative exponent gives a small positive output and why the function still shows growth. For problem 2, name the decay factor and say why the mass never reaches 0. For problem 3, explain why the factor is 0.75 and not 0.25, working one year at a time. For problem 4, compare your balance with the simple-interest balance after two years and explain where the difference comes from.

Show your arithmetic. A reader should be able to reproduce every number you report from the values above without guessing which values or steps you used.

Explain it to a non-expert

A younger cousin has opened a first savings account and asks why the bank keeps talking about "compound" interest. Write a 3–5 sentence message: why compound interest grows faster than simple interest, what multiplying by the same growth factor each year does, and one honest limit or caveat, such as a rate that can change or a gap that is small in the first years. Explain any technical term in everyday words. More than one good caveat can be defended.

For a different, ungraded example, imagine a sourdough starter that doubles in size every 8 hours. Model answer: "The starter does not add the same amount each time; it doubles whatever is already there. So after three doublings it is 2 · 2 · 2 = 8 times as big, not 3 times as big. That is why a baker watches it more closely as the day goes on. One caveat: the doubling only lasts while the starter has fresh flour, so it cannot go on forever."

Use that pattern for the savings account, with your own explanation.

Deliverables

Submit a single exponential report containing:

• the four answers, each with the arithmetic shown and a brief interpretation of what that value shows and one use or limit of it;
• your message for your cousin, explaining why compound interest grows faster than simple interest, what the growth factor does, and one honest caveat; and
• a one-line restatement of the functions and models you used, so your results are reproducible.

Keep it to roughly one page. Clear and correct beats long.

Choose how you show it. Present your report as written text, or as a recorded audio or video explanation of about three minutes with a transcript, or as a labelled sketch with a short note. Every form is graded by the same scoring criteria. Submit in any accessible file type or as a link; a recorded response may be submitted as audio or video with a transcript, or as text.

How you'll be graded

The full analytic rubric is attached to this assignment in Canvas.

Academic integrity and AI use

This version is designed to be completed without AI: do the computations, the interpretations, and the writing yourself, without a chatbot or other AI tool. You may use a calculator or a free graphing calculator such as Desmos to check your arithmetic and your answers. Submitting AI-generated work as your own, or copying another student's numbers, is an integrity violation under the course policy.

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