Textbook Question
In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
y = xe^x-e^x
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 7.3.135
135. Find the area of the “triangular” region in the first quadrant that is bounded above by the curve y = e^(2x), below by the curve y = e^x, and on the right by the line x = ln(3).
Verified step by step guidance1
Identify the region bounded by the curves and the line: the area lies between y = e^{2x} (upper curve) and y = e^{x} (lower curve), from x = 0 (since it's in the first quadrant) to x = \(\ln\)(3) (right boundary).
Set up the integral for the area by subtracting the lower function from the upper function over the interval: the area A is given by the integral \( A = \int_{0}^{\ln(3)} \left(e^{2x} - e^{x}\right) \, dx \).
Recall the integral formulas for exponential functions: \( \int e^{ax} \, dx = \frac{1}{a} e^{ax} + C \), where a is a constant.
Integrate each term separately: compute \( \int e^{2x} \, dx = \frac{1}{2} e^{2x} \) and \( \int e^{x} \, dx = e^{x} \).
Evaluate the definite integral by substituting the limits x = 0 and x = \(\ln\)(3) into the integrated expression \( \left[ \frac{1}{2} e^{2x} - e^{x} \right]_{0}^{\ln(3)} \) and then find the difference to express the area.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definite Integrals for Area Calculation
Definite integrals are used to find the area under a curve between two points. When calculating the area between two curves, the integral of the difference of the functions over the given interval gives the enclosed area.
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Definition of the Definite Integral
Exponential Functions and Their Properties
Exponential functions like y = e^x and y = e^(2x) grow at rates proportional to their current value. Understanding their behavior and how to manipulate their expressions is essential for setting up the integral limits and integrand correctly.
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Determining Intersection Points and Boundaries
Identifying the region's boundaries involves finding where curves intersect and the limits of integration. Here, the vertical boundary x = ln(3) and the first quadrant restriction define the integration interval and ensure the area is correctly bounded.
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6:02Determining Different Coordinates for the Same Point
Related Practice
Textbook Question
Solve the differential equation in Exercises 9–22.
10. (dy/dx) = x²√y, y > 0
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Textbook Question
Which of the functions graphed in Exercises 1–6 are one-to-one, and which are not?
Textbook Question
132. What is special about the functions
f(x) = arcsin((1/√(x²+1)) and g(x)=arctan(1/x)?
Explain.
Textbook Question
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
35. lim (x → 0⁺) ln(x² + 2x) / ln x
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Textbook Question
Suppose that the range of g lies in the domain of f so that the composition fog is defined. If f and g are one-to-one, can anything be said about fog? Give reasons for your answer.