Domanda del libro di testo
80. Find all values of c that satisfy the conclusion of Cauchy's Mean Value Theorem for the given functions and interval.
b. f(x) = x, g(x) = x², (a, b) arbitrary
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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 7.3.2b
In Exercises 1–4, solve for t.
2. b. e^(kt) = 10
Guida verificata passo dopo passo1
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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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.
Video consigliato:
5:47Solving Exponential Equations Using Logs
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