Domanda del libro di testo
22–36. Derivatives Find the derivatives of the following functions.
f(x) = csch⁻¹(2/x)
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Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
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Briggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblema 7.1.3
Briggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblema 7.1.3Capitolo 7, Problema 7.1.3
Evaluate ∫ 4ˣ dx.
Guida verificata passo dopo passo1
Recognize that the integral involves an exponential function with base 4, which can be written as \$4^x$.
Recall the general formula for integrating an exponential function with base \(a\): \(\int a^x \, dx = \frac{a^x}{\ln(a)} + C\), where \(a > 0\) and \(a \neq 1\).
Apply this formula to the integral \(\int 4^x \, dx\), identifying \(a = 4\).
Write the integral as \(\int 4^x \, dx = \frac{4^x}{\ln(4)} + C\), where \(C\) is the constant of integration.
Conclude that the integral is expressed in terms of the exponential function divided by the natural logarithm of the base, plus the constant of integration.
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Exponential Functions
An exponential function has the form a^x, where the base a is a positive constant. Understanding how these functions behave and their properties is essential for integrating expressions like 4^x.
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6:13
Exponential Functions
Integration of Exponential Functions
The integral of a^x with respect to x is (a^x) / ln(a) + C, where a > 0 and a ≠ 1. This formula is derived using substitution and the natural logarithm, and it is key to solving integrals involving exponential terms.
Video consigliato:
05:11Integrals of General Exponential Functions
Natural Logarithm (ln)
The natural logarithm ln(x) is the inverse of the exponential function e^x. It appears in the integration formula for a^x because the derivative of a^x involves ln(a), making it crucial for correctly evaluating the integral.
Video consigliato:
05:18Derivative of the Natural Logarithmic Function
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Bounds on e Use a left Riemann sum with at least n = 2 subintervals of equal length to approximate ln 2 = ∫[1 to 2] (dt/t) and show that ln 2 < 1. Use a right Riemann sum with n = 7 subintervals of equal length to approximate ln 3 = ∫[1 to 3] (dt/t) and show that ln 3 > 1.
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Sketch the graphs of y = cosh x, y = sinh x, and y = tanh x (include asymptotes), and state whether each function is even, odd, or neither.
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Domanda del libro di testo
22–36. Derivatives Find the derivatives of the following functions.
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On what interval is the formula d/dx (tanh⁻¹ x) = 1/(1 - x²) valid?
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Domanda del libro di testo
Evaluate the following derivatives.
d/dx (x^{x¹⁰})
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