Textbook Question
Evaluate the integrals in Exercises 87–96.
89. ∫₀¹ 2^(−θ) dθ
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 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
Chapter 7, Problem 7.1.1
Which of the functions graphed in Exercises 1–6 are one-to-one, and which are not?
Verified step by step guidance1
Step 1: Understand the concept of a one-to-one function. A function is one-to-one if each value of the output (y) corresponds to exactly one value of the input (x). This means the function passes the Horizontal Line Test: no horizontal line intersects the graph more than once.
Step 2: Analyze the given graph of the function \(y = -3x^{3}\). This is a cubic function with a negative leading coefficient, which means it is a decreasing cubic curve passing through the origin.
Step 3: Observe the shape of the graph. Since it is a strictly decreasing curve (it continuously goes down as x increases), it never takes the same y-value twice for different x-values.
Step 4: Apply the Horizontal Line Test. Any horizontal line drawn will intersect the graph at most once, confirming the function is one-to-one.
Step 5: Conclude that the function \(y = -3x^{3}\) is one-to-one because it passes the Horizontal Line Test and each y-value corresponds to exactly one x-value.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
One-to-One Function
A function is one-to-one if each output value corresponds to exactly one input value. This means the function passes the Horizontal Line Test, where no horizontal line intersects the graph more than once. One-to-one functions have inverses that are also functions.
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One-Sided Limits
Horizontal Line Test
The Horizontal Line Test is a graphical method to determine if a function is one-to-one. If any horizontal line crosses the graph more than once, the function is not one-to-one. This test helps verify if the function has a unique inverse.
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05:13Slopes of Tangent Lines
Cubic Functions and Their Graphs
Cubic functions like y = -3x³ have an S-shaped curve and are continuous and smooth. The function y = -3x³ is strictly decreasing, meaning it passes the Horizontal Line Test and is one-to-one. Understanding the shape and behavior of cubic functions aids in analyzing their properties.
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5:53Graph of Sine and Cosine Function
Related Practice
Textbook Question
Each of Exercises 19–24 gives a formula for a function y=f(x) and shows the graphs of f and f^(-1). Find a formula for f^(-1) in each case.
f(x)=x²+1, x≥0
Textbook Question
"In Exercises 59–86, find the derivative of y with respect to the given independent variable.
63. y = x^π"
Textbook Question
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
43. y=√(arcsin x)
Textbook Question
"In Exercises 59–86, find the derivative of y with respect to the given independent variable.
67. y = 7^(sec θ) ln 7"
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