Domanda del libro di testo
6. Which of the following functions grow faster than ln(x) as x→∞? Which grow at the same rate as ln(x)? Which grow slower?
a. log_2(x²)
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Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 7, Problema 7.3.1a
In Exercises 1–4, solve for t.
1. a. e^(-0.3t) = 27
Guida verificata passo dopo passo1
Identify the equation given: \(e^{-0.3t} = 27\).
To solve for \(t\), take the natural logarithm (ln) of both sides to undo the exponential function: \(\ln\left(e^{-0.3t}\right) = \ln(27)\).
Use the logarithm property \(\ln\left(e^x\right) = x\) to simplify the left side: \(-0.3t = \ln(27)\).
Isolate \(t\) by dividing both sides of the equation by \(-0.3\): \(t = \frac{\ln(27)}{-0.3}\).
At this point, you can evaluate \(\ln(27)\) using a calculator and then perform the division to find the value of \(t\).
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Exponential Functions
An exponential function has the form f(t) = a^t, where the variable is in the exponent. Understanding how to manipulate and interpret these functions is essential for solving equations like e^(-0.3t) = 27, where the unknown appears as an exponent.
Video consigliato:
6:13
Exponential Functions
Natural Logarithm (ln)
The natural logarithm is the inverse of the exponential function with base e. Applying the natural logarithm to both sides of an equation like e^(-0.3t) = 27 allows us to 'bring down' the exponent and solve for t.
Video consigliato:
05:18Derivative of the Natural Logarithmic Function
Solving Exponential Equations
Solving exponential equations involves isolating the exponential term and then using logarithms to solve for the variable in the exponent. This process often requires properties of logarithms and careful algebraic manipulation.
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5:47Solving Exponential Equations Using Logs
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Domanda del libro di testo
31. The incidence of a disease (Continuation of Example 4.) Suppose that in any given year the number of cases can be reduced by 25% instead of 20%.
a. How long will it take to reduce the number of cases to 1000?
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Domanda del libro di testo
Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
1. a. arctan 1
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Domanda del libro di testo
153. The linearization of 2ˣ
a. Find the linearization of f(x) = 2ˣ at x = 0. Then round its coefficients to two decimal places.
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Domanda del libro di testo
[Technology Exercise] In Exercises 139–141, find the domain and range of each composite function. Then graph the compositions on separate screens. Do the graphs make sense in each case? Give reasons for your answers. Comment on any differences you see.
141. a. y=arccos(cos x)
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Domanda del libro di testo
143.
a. Show that ∫ ln(x) dx = x ln(x) − x + C.
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