Textbook Question
82. Use the definitions of the hyperbolic functions to find each of the following limits.
a. lim(x→∞) tanh x
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Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- 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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.3.1a
In Exercises 1–4, solve for t.
1. a. e^(-0.3t) = 27
Verified step by step guidance1
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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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
Recommended video:
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.
Recommended video:
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.
Recommended video:
5:47Solving Exponential Equations Using Logs
Related Practice
Textbook Question
Evaluate the integrals in Exercises 67–74 in terms of
a. inverse hyperbolic functions.
71. ∫(from 1/5 to 3/13)dx/(x√(1-16x²))
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Textbook Question
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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Textbook Question
75. a. Find the open intervals on which the function is increasing and decreasing.
g(x) = x(ln x)²
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Textbook Question
145. The linearization of eˣ at x = 0
a. Derive the linear approximation eˣ ≈ 1 + x at x = 0.
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Textbook Question
[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.
139. a. y=arctan(tan x)
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