Domanda del libro di testo
Suppose that the function f and its derivative with respect to x have the following values at x=0, 1, 2, 3, and 4.
Assuming the inverse function f^(-1) is differentiable, find the slope of f^(-1)(x) at
c. x=3
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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 7.3.1c
In Exercises 1–4, solve for t.
1. c. e^((ln 0.2)t) = 0.4
Guida verificata passo dopo passo1
Recognize that the equation is given as \(e^{(\ln 0.2) t} = 0.4\). The goal is to solve for \(t\).
Recall the property of exponents and logarithms: \(e^{\ln a} = a\). This means the expression \(e^{(\ln 0.2) t}\) can be rewritten as \((e^{\ln 0.2})^t = (0.2)^t\).
Rewrite the equation using this property: \((0.2)^t = 0.4\).
To solve for \(t\), take the natural logarithm of both sides: \(\ln((0.2)^t) = \ln(0.4)\).
Use the logarithm power rule to bring down the exponent: \(t \cdot \ln(0.2) = \ln(0.4)\). Then isolate \(t\) by dividing both sides by \(\ln(0.2)\): \(t = \frac{\ln(0.4)}{\ln(0.2)}\).
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2mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Properties of Logarithms and Exponents
Understanding how logarithms and exponents interact is essential, especially that e^(ln a) = a. This allows simplification of expressions like e^((ln 0.2)t) to (0.2)^t, making the equation easier to solve.
Video consigliato:
05:36
Change of Base Property
Solving Exponential Equations
Solving equations where the variable is in the exponent involves rewriting the equation in a comparable base or applying logarithms to isolate the variable. This process transforms the problem into a linear equation in terms of the variable.
Video consigliato:
5:47Solving Exponential Equations Using Logs
Using Natural Logarithms to Isolate Variables
Applying the natural logarithm (ln) to both sides of an equation helps isolate the variable when it appears as an exponent. Since ln and e are inverse functions, this step simplifies the equation and allows solving for the unknown.
Video consigliato:
05:18Derivative of the Natural Logarithmic Function
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