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Questions and Answers
What defines an exponential function?
What defines an exponential function?
In exponential growth, the rate of change is independent of the current value.
In exponential growth, the rate of change is independent of the current value.
False
What is the equation for an exponential function exhibiting doubling?
What is the equation for an exponential function exhibiting doubling?
y = y0 × 2^x
When the exponent is negative, the function shows __________ as x increases.
When the exponent is negative, the function shows __________ as x increases.
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Match the following exponential function features with their descriptions:
Match the following exponential function features with their descriptions:
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What is the doubling time of the bacteria from t=0 to t=6 hours?
What is the doubling time of the bacteria from t=0 to t=6 hours?
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If a population of bacteria doubles every 8 hours and starts with 5000 bacteria, how many will there be after 32 hours?
If a population of bacteria doubles every 8 hours and starts with 5000 bacteria, how many will there be after 32 hours?
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Exponential decay can be modeled by the equation y = 100 × 2^-x.
Exponential decay can be modeled by the equation y = 100 × 2^-x.
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After 24 hours, there will be 28 times the initial number of bacteria.
After 24 hours, there will be 28 times the initial number of bacteria.
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What is the value of q when the maximum charge Q is 10 μC and the time constant is 5 ms, if q = 8 μC?
What is the value of q when the maximum charge Q is 10 μC and the time constant is 5 ms, if q = 8 μC?
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How many radioactive nuclei are left after 200 days if the half-life is 80 days and the initial amount is 1,000,000?
How many radioactive nuclei are left after 200 days if the half-life is 80 days and the initial amount is 1,000,000?
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The formula for the charge of a capacitor is $q = Q(1 - e^{-\frac{t}{\tau}})$, where Q is the maximum charge and τ is the _____.
The formula for the charge of a capacitor is $q = Q(1 - e^{-\frac{t}{\tau}})$, where Q is the maximum charge and τ is the _____.
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Match the physical examples with their corresponding concepts:
Match the physical examples with their corresponding concepts:
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How many bacteria will be present at t=24 hours?
How many bacteria will be present at t=24 hours?
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The formula to calculate distance in motion is $x = 0.5 × (acceleration) × (time)^2$.
The formula to calculate distance in motion is $x = 0.5 × (acceleration) × (time)^2$.
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If a trooper accelerates at 3 m/s² for 16.92 seconds, what distance does he cover?
If a trooper accelerates at 3 m/s² for 16.92 seconds, what distance does he cover?
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What is the maximum charge (Q) when charging a capacitor in the given example?
What is the maximum charge (Q) when charging a capacitor in the given example?
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When t = 0, the charge (q) on a capacitor is equal to the maximum charge (Q).
When t = 0, the charge (q) on a capacitor is equal to the maximum charge (Q).
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What fraction of X-ray intensity is transmitted through a 0.5 mm thick plate with an attenuation coefficient of 7 mm⁻¹?
What fraction of X-ray intensity is transmitted through a 0.5 mm thick plate with an attenuation coefficient of 7 mm⁻¹?
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The doubling time of bacteria is _____ hours.
The doubling time of bacteria is _____ hours.
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Match the following variables with their meanings:
Match the following variables with their meanings:
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How long will it take for a capacitor to reach 63.2% of its maximum charge?
How long will it take for a capacitor to reach 63.2% of its maximum charge?
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To absorb 50% of the entering intensity, the thickness of the plate should be approximately 0.099 mm.
To absorb 50% of the entering intensity, the thickness of the plate should be approximately 0.099 mm.
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What is the value of $y$ when $x = 200$ days, $y_0 = 106$, and $T = 80$ days in the equation $y = y_0 imes 2^{-x/T}$?
What is the value of $y$ when $x = 200$ days, $y_0 = 106$, and $T = 80$ days in the equation $y = y_0 imes 2^{-x/T}$?
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Study Notes
Exponential Functions
- Exponential functions have a variable in the exponent and a constant base.
- Examples include: y = 2x (base = 2), f(x) = ex (base = e).
- Growth or decay is proportional to the function's current value. Large values yield a large rate of change; small values yield a small rate of change.
- The greater the starting value, the greater the change.
- If x increases by 1, y gets multiplied by the base.
- A basic exponential function looks like: y = y0 × bx
Exponential Growth
- The number of bacteria in a Petri dish at t=0 is 5 × 104.
- Doubling time is 8 hours.
- How many bacteria will there be at t=64 hours?
- Calculation: y = 5 × 104 × 264/8 = 1.28 × 107
Exponential Decay
- Half-life of a radioactive sample is 80 days.
- Initial number of radioactive nuclei is 106.
- How many will be left in 200 days?
- Calculation: y = 106 × 2−200/80 = 1.77 × 105
Exponential Rise to Maximum
- A negative exponential is subtracted from a constant (maximum value).
- Example: y = y0 × (1 − e−x/τ).
- In this example, y0 = 10, is the maximum value.
- Physical example: charging a capacitor.
- The time constant (τ) characterizes the speed of charging; smaller τ means faster charging
Exponential Decay (Negative Exponential)
- When the exponent is negative, y decreases as x increases.
- Example: y = 100 × 2−x.
Motion in One Dimension
- Displacement: The change in position of an object along the x-axis, given as ∆x = xf - xi.
- Velocity: Average speed is path length divided by elapsed time. Average velocity is displacement divided by time interval. Instantaneous velocity is the slope of the position-time graph at a given instant.
- Acceleration: Average acceleration is the change in velocity divided by the time interval. Instantaneous acceleration is the slope of the velocity-time graph at a given instant.
- One-Dimensional Motion with Constant Acceleration: Equations describing motion with constant acceleration include v = v0 + at, ∆x = v0t + ½at2 and v2 = v02 + 2a∆x.
- Freely Falling Objects: Objects falling under gravity have constant downward acceleration (≈9.8 m/s2 in the absence of air resistance)
Additional Notes (Various Problems)
- Attenuation coefficient of a metal: If the attenuation coefficient of a metal is 7 mm−1, what fraction of X-ray intensity will be absorbed by a 0.5 mm thick plate? Calculation: I/I0 = e−µx
- Bacteria doubling time: The number of bacteria in a Petri dish at t = 0 is 6 × 105 and 6 hours later there are 2.4 × 106. What is the doubling time? 3 hours.
- Charging a capacitor: If the time constant is 5 ms, after how much time is q = 8µC? Calculation: q = Q(1 − e−t/τ). t≈8.05ms
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Description
Explore the fundamentals of exponential functions, including growth and decay. This quiz covers key concepts, examples, and practical applications such as bacterial growth and radioactive decay calculations. Test your understanding of how these functions behave and apply mathematical reasoning to real-world scenarios.
