Domanda del libro di testo
13-26 Implicit differentiation Carry out the following steps.
b. Find the slope of the curve at the given point.
tan xy = x+y; (0,0)
184
views
Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
TOC.tableOfContents
- 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: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 3, Problema 3.9.56b
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 dE/dM and evaluate it for M=3. What does this derivative mean? (M has no units, so the units of the derivative are J per change in magnitude.)
Guida verificata passo dopo passo1
First, identify the function given in the problem: \( E = 25,000 \cdot 10^{1.5M} \). This represents the energy released by an earthquake as a function of its magnitude \( M \).
To find \( \frac{dE}{dM} \), we need to differentiate the function \( E \) with respect to \( M \). Notice that the function involves an exponential term \( 10^{1.5M} \).
Apply the chain rule for differentiation. The derivative of \( 10^{1.5M} \) with respect to \( M \) is \( 10^{1.5M} \cdot \ln(10) \cdot 1.5 \). Multiply this by the constant 25,000 to get the derivative: \( \frac{dE}{dM} = 25,000 \cdot 10^{1.5M} \cdot \ln(10) \cdot 1.5 \).
Now, substitute \( M = 3 \) into the derivative \( \frac{dE}{dM} \) to evaluate it at this specific magnitude. This involves calculating \( 25,000 \cdot 10^{4.5} \cdot \ln(10) \cdot 1.5 \).
The derivative \( \frac{dE}{dM} \) represents the rate of change of energy with respect to the magnitude of the earthquake. It tells us how much the energy released by the earthquake increases for a small increase in magnitude \( M \). The units of this derivative are joules per change in magnitude.
Risposta video verificata per un problema simile:
Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
5mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Derivative
A derivative represents the rate of change of a function with respect to a variable. In this context, dE/dM indicates how the energy E released by an earthquake changes as the magnitude M changes. It quantifies the sensitivity of energy release to variations in magnitude, providing insight into the relationship between these two variables.
Video consigliato:
Percorso guidato
05:44
Derivatives
Exponential Functions
The equation E=25,000 ⋅ 10<sup>1.5M</sup> is an example of an exponential function, where the variable M is in the exponent. Exponential functions grow rapidly, and in this case, they illustrate how small increases in magnitude can lead to significant increases in energy release. Understanding the properties of exponential functions is crucial for analyzing the behavior of E as M changes.
Video consigliato:
6:13Exponential Functions
Magnitude Scale
The magnitude scale, originally developed by Charles Richter, quantifies the size of earthquakes. It is a logarithmic scale, meaning each whole number increase on the 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 derivative, as it contextualizes how energy release varies with changes in magnitude.
Video consigliato:
Percorso guidato
06:21Properties of Functions
Pratica correlata
Domanda del libro di testo
Derivatives using tables Let and . Use the table to compute the following derivatives.
<IMAGE>
b.
278
views
Domanda del libro di testo
{Use of Tech} Angle of elevation A small plane, moving at 70 m/s, flies horizontally on a line 400 meters directly above an observer. Let θ be the angle of elevation of the plane (see figure). <IMAGE>
b. Graph dθ/dx as a function of x and determine the point at which θ changes most rapidly.
160
views
Domanda del libro di testo
Derivatives and tangent lines
b. Determine an equation of the line tangent to the graph of f at the point (a,f(a)) for the given value of a.
f(x) = √3x; a= 12
227
views
Domanda del libro di testo
A differential equation is an equation involving an unknown function and its derivatives. Consider the differential equation y′′(t)+y(t) = 0.
b. Show that y = B cos t satisfies the equation for any constant B.
254
views
Domanda del libro di testo
109-112 {Use of Tech} Calculating limits The following limits are the derivatives of a composite function g at a point a.
b. Use the Chain Rule to find each limit. Verify your answer by using a calculator.
242
views