Textbook Question
82. For what values of a and b is
lim(x→0)(tan(2x/x³) + a/x² + sin(bx)/x) = 0?
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Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- 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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.3.25
In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
y = ∫(from 0 to lnx) sin(e^t) dt
Verified step by step guidance1
Identify the function given: \( y = \int_0^{\ln x} \sin(e^t) \, dt \). This is a definite integral with a variable upper limit \( \ln x \).
Recall the Fundamental Theorem of Calculus Part 1, which states that if \( y = \int_a^{g(x)} f(t) \, dt \), then \( \frac{dy}{dx} = f(g(x)) \cdot g'(x) \).
In this problem, \( f(t) = \sin(e^t) \) and the upper limit is \( g(x) = \ln x \). The lower limit is a constant (0), so it does not affect the derivative.
Compute the derivative of the upper limit: \( \frac{d}{dx} (\ln x) = \frac{1}{x} \).
Apply the chain rule: \( \frac{dy}{dx} = \sin(e^{\ln x}) \cdot \frac{1}{x} \). Note that \( e^{\ln x} = x \), so the derivative simplifies to \( \frac{dy}{dx} = \sin(x) \cdot \frac{1}{x} \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Fundamental Theorem of Calculus
This theorem connects differentiation and integration, stating that if a function is defined as an integral with a variable upper limit, its derivative is the integrand evaluated at that limit. It allows us to differentiate an integral with variable limits directly.
Recommended video:
06:11
Fundamental Theorem of Calculus Part 1
Chain Rule
The chain rule is used to differentiate composite functions. When the upper limit of the integral is a function of x (like ln(x)), we must multiply the derivative of the integral's upper limit by the derivative of that function to find the overall derivative.
Recommended video:
05:02Intro to the Chain Rule
Properties of Logarithmic and Exponential Functions
Understanding the derivatives of ln(x) and e^t is essential here. The derivative of ln(x) is 1/x, and e^t is its own derivative. These properties help simplify the expression when applying the chain rule and evaluating the integrand.
Recommended video:
06:21Properties of Functions
Related Practice
Textbook Question
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
65. y = (cos θ)^(√2)
Textbook Question
44. Silver cooling in air The temperature of an ingot of silver is 60°C above room temperature right now. Twenty minutes ago, it was 70°C above room temperature. How far above room temperature will the silver be
b. 2 hours from now?
Textbook Question
Solve the differential equation in Exercises 9–22.
17. (dy/dx) = 2x(y - 1), y > 1
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Textbook Question
In Exercises 25–36, find the derivative of y with respect to the appropriate variable.
35. y = sinh⁻¹(tan x)
Textbook Question
Evaluate the integrals in Exercises 87–96.
93. ∫₀^(π/2) 7^(cos t) sin t dt