Textbook Question
Calculate the derivative of the following functions.
y = (1 - e0.05x)-1
Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 3, Problem 56a
The energy (in joules) released by an earthquake of magnitude M is given by the equation E = 25,000 ⋅ 101.5M. (This equation can be solved for M to define the magnitude of a given earthquake; it is a refinement of the original Richter scale created by Charles Richter in 1935.)
Compute the energy released by earthquakes of magnitude 1, 2, 3, 4, and 5. Plot the points on a graph and join them with a smooth curve.
Verified step by step guidance1
Understand the given formula for energy released by an earthquake: E = 25,000 ⋅ 10^(1.5M). This formula relates the magnitude M of an earthquake to the energy E it releases.
To compute the energy for each magnitude, substitute the values of M (1, 2, 3, 4, and 5) into the formula one by one. For example, for M = 1, calculate E = 25,000 ⋅ 10^(1.5 * 1).
Repeat the substitution for each magnitude: M = 2, M = 3, M = 4, and M = 5, calculating the corresponding energy E for each case using the formula E = 25,000 ⋅ 10^(1.5M).
Once you have calculated the energy values for each magnitude, plot these points on a graph with the magnitude M on the x-axis and the energy E on the y-axis.
After plotting the points, draw a smooth curve through them to visualize how the energy released by an earthquake increases with its magnitude. This curve will help illustrate the exponential relationship between magnitude and energy.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Exponential Functions
The equation E = 25,000 ⋅ 10^(1.5M) is an example of an exponential function, where the variable M (magnitude) affects the energy E exponentially. In exponential functions, a constant base is raised to a variable exponent, leading to rapid growth or decay. Understanding how to manipulate and evaluate exponential functions is crucial for calculating energy values for different magnitudes.
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Exponential Functions
Magnitude Scale
The magnitude of an earthquake, represented by M, is a logarithmic scale that quantifies the size of seismic events. Each whole number increase on the Richter scale corresponds to a tenfold increase in measured amplitude and approximately 31.6 times more energy release. This concept is essential for interpreting the results of the energy calculations and understanding the relationship between magnitude and energy.
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Graphing Data
Graphing involves plotting calculated values on a coordinate system to visualize relationships between variables. In this case, the magnitudes (M) will be plotted on the x-axis and the corresponding energy values (E) on the y-axis. Understanding how to create and interpret graphs is vital for analyzing the results and observing trends, such as the smooth curve that represents the relationship between magnitude and energy.
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Related Practice
Textbook Question
Find and simplify the derivative of the following functions.
h(x) = (5x7 + 5x)(6x3 + 3x2 + 3)
Textbook Question
Power and energy are often used interchangeably, but they are quite different. Energy is what makes matter move or heat up. It is measured in units of joules or Calories, where 1 Cal=4184 J. One hour of walking consumes roughly 10⁶J, or 240 Cal. On the other hand, power is the rate at which energy is used, which is measured in watts, where 1 W = 1 J/s. Other useful units of power are kilowatts (1 kW=10³ W) and megawatts (1 MW=10⁶ W). If energy is used at a rate of 1 kW for one hour, the total amount of energy used is 1 kilowatt-hour (1 kWh = 3.6×10⁶ J) Suppose the cumulative energy used in a large building over a 24-hr period is given by E(t)=100t + 4t² − (t³ / 9) kWh where t = 0 corresponds to midnight.
The power is the rate of energy consumption; that is, P(t) = E′(t) Find the power over the interval 0 ≤ t ≤ 24.
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Textbook Question
An object oscillates along a vertical line, and its position in centimeters is given by y(t)=30(sint - 1), where t ≥ 0 is measured in seconds and y is positive in the upward direction.
At what times and positions is the velocity zero?
Textbook Question
An object oscillates along a vertical line, and its position in centimeters is given by y(t) = 30(sin t - 1), where t ≥ 0 is measured in seconds and y is positive in the upward direction.
The acceleration of the oscillator is a(t) = v′(t). Find and graph the acceleration function.
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Textbook Question
Calculate the derivative of the following functions.
y = cos7/4(4x3)