Curriculum / Math / 11th Grade / Unit 5: Exponential Modeling and Logarithms / Lesson 4
Math
Unit 5
11th Grade
Lesson 4 of 16
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Define and use $$e$$ in continuous compounding situations.
The core standards covered in this lesson
A.SSE.B.3.C — Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15<sup>t</sup> can be rewritten as (1.151/12)<sup>12t</sup> 1.012<sup>12t</sup> to reveal the approximate equivalent monhly interest rate if the annual rate is 15%.
F.BF.A.1.A — Determine an explicit expression, a recursive process, or steps for calculation from a context.
The essential concepts students need to demonstrate or understand to achieve the lesson objective
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Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding
A man is investing $100 at (the absurd) interest rate of 100% interest. He is exploring how the rate of annual return changes as the rate of compounding, $$n$$, becomes more and more frequent. Write a table of values that shows the growth rate for $$\left ( 1+{1\over n} \right )^{nt}$$ as $$n$$ approaches infinity.
Compounding Interest with a 100% Interest Rate, accessed on Feb. 22, 2018, 3:03 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.
Four physicists describe the amount of a radioactive substance, $$Q$$ in grams, left after $$t$$ years:
a. $$Q=300e^{-0.0577t}$$
b. $$Q=300(1/2)^{t/12}$$
c. $$Q=300 \cdot 0.9439^t$$
d. $$Q=252.290 \cdot 0.9439^{t-3}$$
Forms of Exponential Expressions, accessed on Feb. 22, 2018, 3:04 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.
A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved
Suppose $5,000 is put into an account that pays 4% compounded continuously. How much will be in the account after 3 years?
The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.
Lesson 3
Lesson 5
Topic A: Modeling with and Interpreting Exponential Functions
Identify, model, and analyze geometric sequences.
F.IF.A.3 F.IF.B.5 F.LE.A.2
Analyze and construct exponential functions that model contexts.
F.IF.B.4 F.IF.C.8.B F.LE.A.2
Write and change the form of exponential functions that model compounding interest.
F.BF.A.1.A F.LE.B.5
A.SSE.B.3.C F.BF.A.1.A
Describe the derivation of the formula for the sum of a finite geometric series and use it to solve problems.
A.SSE.B.4
Find the sum of an infinite geometric series.
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Topic B: Definition and Meaning of Logarithms
Describe and evaluate simple numeric logarithms (Part I).
F.LE.A.4
Describe and evaluate simple numeric logarithms (Part II).
Describe logarithms as the inverse of exponential functions and graph logarithmic functions.
F.BF.B.3 F.BF.B.4.B F.BF.B.4.C F.BF.B.5 F.IF.C.7.E
Evaluate common and natural logs using tables, graphs, and calculators.
F.BF.B.4.C F.LE.A.4
Understand and apply the change of base property to evaluate logarithms.
Develop and use the product and quotient properties of logarithms to write equivalent expressions.
Develop and use the power property of logarithms to write equivalent expressions.
F.BF.B.4.B F.LE.A.4
Solve equations with logarithms.
Use logarithms to solve exponential modeling problems (Part I).
A.SSE.A.1.B F.LE.A.4
Use logarithms to solve exponential modeling problems (Part II).
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