Textbook Question
91. [Technology Exercise] 91. The continuous extension of to (sin x)^x to [0, π]
b. Verify your conclusion in part (a) by finding lim(x→0⁺)f(x) with l’Hôpital’s Rule.
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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 7.3.2b
In Exercises 1–4, solve for t.
2. b. e^(kt) = 10
Verified step by step guidance1
Identify the equation given: \(e^{kt} = 10\).
To solve for \(t\), take the natural logarithm (ln) of both sides to utilize the property that \(\ln(e^x) = x\).
Apply the natural logarithm: \(\ln(e^{kt}) = \ln(10)\).
Simplify the left side using the logarithm property: \(kt = \ln(10)\).
Isolate \(t\) by dividing both sides by \(k\): \(t = \frac{\ln(10)}{k}\).
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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. In this problem, e^(kt) represents an exponential function with base e, the natural exponential constant approximately equal to 2.718. Understanding how exponential functions behave is essential for solving equations involving them.
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6:13
Exponential Functions
Natural Logarithm (ln)
The natural logarithm is the inverse function of the exponential function with base e. It allows us to solve equations where the variable is in the exponent by 'undoing' the exponential. Applying ln to both sides of e^(kt) = 10 helps isolate the variable t.
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05:18Derivative of the Natural Logarithmic Function
Solving Exponential Equations
To solve equations like e^(kt) = 10, we use logarithms to isolate the exponent. After taking the natural logarithm of both sides, we apply algebraic manipulation to solve for t, including dividing by the constant k. This process is fundamental in calculus and related fields.
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5:47Solving Exponential Equations Using Logs
Related Practice
Textbook Question
2. Express the following logarithms in terms of ln 5 and ln 7.
b. ln 9.8
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Textbook Question
In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:
b. Solve the equation y=f(x) for x as a function of y, and name the resulting inverse function g.
67. y= √(3x-2), 2/3 ≤ x ≤ 4, x_0=3
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Textbook Question
1. Express the following logarithms in terms of ln 2 and ln 3.
b. ln(4/9)
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Textbook Question
131. Let f(x) = x * e^(−x).
b. Find all inflection points for f.
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Textbook Question
71. Locate and identify the absolute extreme values of cos(ln x) on [1/2, 2]
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