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Questions and Answers
What is the approximate value of e correct to five decimal places?
What is the approximate value of e correct to five decimal places?
The graph of the function y = e^x crosses the y-axis with a slope of -1.
The graph of the function y = e^x crosses the y-axis with a slope of -1.
False
What is the domain of the function y = e^x?
What is the domain of the function y = e^x?
All real numbers
The range of the function y = e^x is __________.
The range of the function y = e^x is __________.
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Match the following transformations with their effects on the graph of y = e^x:
Match the following transformations with their effects on the graph of y = e^x:
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What is the value of the function f(x) = 2x when x = 0?
What is the value of the function f(x) = 2x when x = 0?
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The graph of y = bx will always decrease if the base b is greater than 1.
The graph of y = bx will always decrease if the base b is greater than 1.
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What is the domain of the function y = 3 - 2x?
What is the domain of the function y = 3 - 2x?
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The range of the function y = 3 - 2x is (_____ , 3).
The range of the function y = 3 - 2x is (_____ , 3).
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Match the exponential function to its characteristic:
Match the exponential function to its characteristic:
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Which of the following describes the graph of y = (1/b)x?
Which of the following describes the graph of y = (1/b)x?
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If b = 1, the exponential function y = bx doesn't change with x.
If b = 1, the exponential function y = bx doesn't change with x.
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How does the value of the base b affect the growth of the function y = bx?
How does the value of the base b affect the growth of the function y = bx?
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What is the initial mass mentioned in the example?
What is the initial mass mentioned in the example?
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The mass remaining after 40 years from the initial value is 5 mg.
The mass remaining after 40 years from the initial value is 5 mg.
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How long will it take for the mass to be reduced to 5 mg?
How long will it take for the mass to be reduced to 5 mg?
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The slope of the tangent line to an exponential graph at the point (0, 1) for the function $y = e^x$ is ______.
The slope of the tangent line to an exponential graph at the point (0, 1) for the function $y = e^x$ is ______.
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Match the following values with their significance:
Match the following values with their significance:
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Which statement describes the behavior of the mass over time?
Which statement describes the behavior of the mass over time?
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The natural exponential function has a slope at the point (0, 1) that is greater than 1.
The natural exponential function has a slope at the point (0, 1) that is greater than 1.
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Who is the mathematician that designated the letter e as the base for the natural exponential function?
Who is the mathematician that designated the letter e as the base for the natural exponential function?
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Study Notes
Exponential Functions
- Exponential functions are functions of the form f(x) = bx, where b is a positive constant.
- When x = n (a positive integer), bn = b × b × ... × b (n factors of b).
- If x = 0, then b0 = 1.
- If x = -n (where n is a positive integer), then b-n = 1/bn.
- For rational numbers x (x = p/q), bx = bp/q = (bp)1/q = q√(bp) = (q√b)p.
- The meaning of bx for irrational x is defined by limiting values of bx , where x approaches an irrational number using better approximations of that number.
- The graph of y = 2x, where x is rational, shows holes. To define f(x) = 2x for all real numbers, fill in these holes with suitable values.
- For irrational numbers √3, 2√3 is determined using approximations; 1.7 < √3 < 1.8, so 21.7< 2√3 < 21.8.
- Exponential functions (y = bx) where b ≠ 1 have domain R (all real numbers) and range (0, ∞)
- Exponential functions with 0<b<1 decrease, b = 1 is a constant, and b > 1 increases.
- Note: Rules of exponents apply to all real numbers (not just rationals). These rules are:
- bx+y = bx × by
- bx-y = bx/by
- (bx)y = bxy
- (ab)x = ax × bx
Applications of Exponential Functions
- Exponential functions are frequently used in mathematical models for processes like population growth and radioactive decay.
- Example: A bacterial population doubles hourly, and p(0) = 1000 (initial population); then p(t) = 1000 × 2t.
- Example: The human population (approximate exponential growth model): P = (1436.53)(1.01395)t where t = 0 corresponds to 1900.
The Number e
- There is one special base (e ≈ 2.71828) for exponential functions which makes certain calculus formulas simpler.
- The slope of the tangent line to y = ex at (0, 1) is exactly 1.
- f(x) = ex is called the natural exponential function.
Examples and Exercises (Summary)
- Example 1: Graphing functions like y = 3 - 2x, by reflecting, shifting, and scaling.
- Example 2: Discusses the half-life of strontium-90 (90Sr) and its exponential decay.
- Numerous exercises, involving laws of exponents, sketching and graph of expressions, and graph transformations are listed to be completed.
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Description
Explore the fascinating world of exponential functions defined by f(x) = b^x. This quiz covers properties such as values for positive, negative, and rational exponents, as well as the significance of their graphs for real numbers. Challenge your understanding of how these functions behave, including their interpretations for irrational numbers.
