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Questions and Answers
What is the formula for the effective annual interest rate when interest is compounded k times per year?
What is the formula for the effective annual interest rate when interest is compounded k times per year?
If $1000 is invested at a nominal annual rate of 1.2% compounded monthly, what is the worth of the investment after 24 months?
If $1000 is invested at a nominal annual rate of 1.2% compounded monthly, what is the worth of the investment after 24 months?
How does the effective interest rate change when interest is compounded weekly compared to monthly?
How does the effective interest rate change when interest is compounded weekly compared to monthly?
What is an essential characteristic of exponential functions in the context of compounding interest?
What is an essential characteristic of exponential functions in the context of compounding interest?
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What happens to an investment if a bank deposits $1 into the account each month compared to compounding the initial investment?
What happens to an investment if a bank deposits $1 into the account each month compared to compounding the initial investment?
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What is the relationship between the number of times interest is compounded and the two formulas regarding future value?
What is the relationship between the number of times interest is compounded and the two formulas regarding future value?
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Which mathematical limit defines Euler’s number, e?
Which mathematical limit defines Euler’s number, e?
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How can you isolate the principal P from the future value formula B(T) when r and k are fixed?
How can you isolate the principal P from the future value formula B(T) when r and k are fixed?
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What is the formula for the present value of an investment with periodic compounding?
What is the formula for the present value of an investment with periodic compounding?
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What does the effective annual interest rate account for in relation to nominal interest?
What does the effective annual interest rate account for in relation to nominal interest?
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In the context of continuous compounding, how is the present value calculated?
In the context of continuous compounding, how is the present value calculated?
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What happens to the two formulas for future value as the frequency of compounding increases infinitely?
What happens to the two formulas for future value as the frequency of compounding increases infinitely?
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What is the approximate value of Euler's number, e?
What is the approximate value of Euler's number, e?
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What defines an exponential function?
What defines an exponential function?
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What is a property of exponential functions for $b > 0$?
What is a property of exponential functions for $b > 0$?
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What happens to the limit of an exponential function as $x$ approaches negative infinity for $b > 1$?
What happens to the limit of an exponential function as $x$ approaches negative infinity for $b > 1$?
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Which of these statements about the future value of an investment is correct?
Which of these statements about the future value of an investment is correct?
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How does an exponential function behave as $x$ approaches positive infinity for $0 < b < 1$?
How does an exponential function behave as $x$ approaches positive infinity for $0 < b < 1$?
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Which of the following is true regarding the multiplication rule for exponential functions?
Which of the following is true regarding the multiplication rule for exponential functions?
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What is the value of $f(x)$ at the y-axis when $b > 0$?
What is the value of $f(x)$ at the y-axis when $b > 0$?
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Which of these statements is true regarding the equality rule for exponential functions?
Which of these statements is true regarding the equality rule for exponential functions?
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Study Notes
Exponential Functions
- Exponential functions are functions of the form bx , where b > 0 and b ≠ 1.
- A function grows exponentially if its growth depends on its current value.
- Exponential functions are continuous (no jumps or breaks).
- bx is always greater than 0 for any x.
- bx intercepts the y-axis at (0, 1).
Properties of Exponential Functions
- For b > 0:
- bx is defined for all real numbers x (i.e., -∞ < x < ∞).
- bx is a continuous function.
- If b > 1, the function exhibits exponential growth.
- If 0 < b < 1, the function exhibits exponential decay.
Limits of Exponential Functions
- For b > 1:
- The limit of bx as x approaches negative infinity is 0 (limx→-∞ bx = 0).
- The limit of bx as x approaches positive infinity is ∞ (limx→∞ bx = ∞).
- For 0 < b < 1:
- The limit of bx as x approaches negative infinity is ∞ (limx→-∞ bx = ∞).
- The limit of bx as x approaches positive infinity is 0 (limx→∞ bx = 0).
Rules for Exponential Functions
- Equality: If b ≠ 1, bx = by if and only if x = y.
- Product: bx * by = bx+y
- Quotient: bx / by = bx-y
- Power: (bx)y = bxy
- Multiplication: (ab)x = ax * bx
- Division: (a/b)x = ax / bx
Future Value of an Investment
-
Compounding k times per year: B(t) = P(1 + r/k)kt, where:
- P is the principal amount (initial investment)
- r is the annual interest rate (as a decimal)
- k is the number of times interest is compounded per year
- t is the number of years
-
Continuous Compounding: B(t) = Pert, where:
- e is Euler's number (approximately 2.718)
Euler's Number
- Euler's number (e) is defined as the limit: limk→∞ (1 + 1/k)k = e ≈ 2.718.
Present Value of an Investment
- Compounding k times per year: P(T) = B(1 + r/k)-kT
- Continuous Compounding: P(T) = Be-rT
Effective Interest
- Effective interest rate (re) reflects the actual amount earned per year, considering the compounding frequency.
- If interest is compounded k times per year at a nominal rate r, then the effective annual interest rate is re = (1 + r/k)k - 1
- If interest is compounded continuously, then the effective annual interest rate is re = er - 1
Applications
- Exponential functions model compound interest.
- Effective interest rates are used to compare interest rates.
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Description
Explore the fundamentals of exponential functions, including their properties, growth and decay, and the limits as x approaches infinity. Understand how these functions behave and their applications in mathematics. Perfect for students looking to grasp the concepts of exponential functions in detail.
