Domanda del libro di testo
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
23. y=arcsin(√2t)
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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 7.3.5
5. e^(2t)-3e^t = 0
Guida verificata passo dopo passo1
Rewrite the equation \(e^{2t} - 3e^t = 0\) by recognizing that \(e^{2t}\) can be expressed as \((e^t)^2\). This allows us to treat the equation like a quadratic in terms of \(e^t\).
Let \(x = e^t\). Substitute this into the equation to get \(x^2 - 3x = 0\).
Factor the quadratic equation: \(x(x - 3) = 0\). This gives two possible solutions for \(x\): \(x = 0\) or \(x = 3\).
Recall that \(x = e^t\), and since \(e^t\) is never zero for any real \(t\), discard \(x = 0\). Focus on \(e^t = 3\).
Solve for \(t\) by taking the natural logarithm of both sides: \(t = \ln(3)\). This gives the solution for \(t\).
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The substitution method replaces a complex expression with a simpler variable to transform the equation into a more familiar form, such as turning e^t into x, making it easier to solve.
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