Domanda del libro di testo
7–28. Derivatives Evaluate the following derivatives.
d/dx (ln (cos² 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 7.53
Briggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblema 7.53Capitolo 7, Problema 7.53
37–56. Integrals Evaluate each integral.
∫ (cosh z) / (sinh² z) dz
Guida verificata passo dopo passo1
Recognize that the integral involves hyperbolic functions: \(\cosh z\) and \(\sinh z\). Recall the derivatives: \(\frac{d}{dz} \sinh z = \cosh z\) and \(\frac{d}{dz} \cosh z = \sinh z\).
Rewrite the integral as \(\int \frac{\cosh z}{\sinh^{2} z} \, dz = \int \cosh z \cdot \sinh^{-2} z \, dz\) to see the structure more clearly.
Use substitution by letting \(u = \sinh z\). Then, \(du = \cosh z \, dz\), which means \(\cosh z \, dz = du\).
Substitute into the integral to get \(\int u^{-2} \, du\), which simplifies the integral to a power function of \(u\).
Integrate \(\int u^{-2} \, du\) using the power rule for integrals, then substitute back \(u = \sinh z\) to express the answer in terms of \(z\).
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Hyperbolic Functions
Hyperbolic functions, such as sinh(z) and cosh(z), are analogs of trigonometric functions but based on exponential functions. They satisfy identities like cosh²(z) - sinh²(z) = 1, which are useful in simplifying expressions and integrals involving these functions.
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Percorso guidato
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Asymptotes of Hyperbolas
Integration Techniques for Rational Functions
Integrals involving ratios of functions often require substitution or rewriting the integrand to a simpler form. Recognizing derivatives within the integrand, such as identifying if the numerator is the derivative of the denominator, helps in applying substitution effectively.
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Substitution Method in Integration
The substitution method involves changing variables to simplify an integral. By letting u equal a function inside the integral (e.g., u = sinh(z)), the integral can be transformed into a more straightforward form, making it easier to evaluate.
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07:33Euler's Method
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27–30. Designing exponential decay functions Devise an exponential decay function that fits the following data; then answer the accompanying questions. Be sure to identify the reference point (t = 0) and units of time.
Valium metabolism The drug Valium is eliminated from the bloodstream with a half-life of 36 hr. Suppose a patient receives an initial dose of 20 mg of Valium at midnight. How much Valium is in the patient’s blood at noon the next day? When will the Valium concentration reach 10% of its initial level?
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Domanda del libro di testo
A calculator has a built-in sinh⁻¹ x function, but no csch⁻¹ x function. How do you evaluate csch⁻¹ 5 on such a calculator?
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Domanda del libro di testo
15–20. Designing exponential growth functions Complete the following steps for the given situation.
a. Find the rate constant k and use it to devise an exponential growth function that fits the given data.
b. Answer the accompanying question.
Cell growth The number of cells in a tumor doubles every 6 weeks starting with 8 cells. After how many weeks does the tumor have 1500 cells?
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Domanda del libro di testo
Sag angle Imagine a climber clipping onto the rope described in Example 7 and pulling himself to the rope’s midpoint. Because the rope is supporting the weight of the climber, it no longer takes the shape of the catenary y = 200 cosh x/200. Instead, the rope (nearly) forms two sides of an isosceles triangle. Compute the sag angle θ illustrated in the figure, assuming the rope does not stretch when weighted. Recall from Example 7 that the length of the rope is 101 ft.
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Domanda del libro di testo
37–56. Integrals Evaluate each integral.
∫ tanh²x dx (Hint: Use an identity.)
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