Textbook Question
What happens if you take B = 2π in the addition formulas? Do the results agree with something you already know?
Ch. 1 - 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 1, Problem 1.2.21
Composition of Functions
A balloon’s volume V is given by V = s² + 2s + 3 cm³, where s is the ambient temperature in °C. The ambient temperature s at time t minutes is given by s = 2t − 3 °C. Write the balloon’s volume V as a function of time t.
Verified step by step guidance1
Identify the given functions: The volume of the balloon is given by \( V = s^2 + 2s + 3 \) and the ambient temperature is given by \( s = 2t - 3 \).
Recognize that you need to find the composition of functions, specifically \( V(s(t)) \), which means substituting the expression for \( s \) into the volume function \( V \).
Substitute \( s = 2t - 3 \) into the volume function \( V = s^2 + 2s + 3 \). This gives \( V = (2t - 3)^2 + 2(2t - 3) + 3 \).
Expand the expression \( (2t - 3)^2 \) using the formula \( (a - b)^2 = a^2 - 2ab + b^2 \). This results in \( 4t^2 - 12t + 9 \).
Combine all terms: \( V = 4t^2 - 12t + 9 + 4t - 6 + 3 \). Simplify the expression to write the volume \( V \) as a function of time \( t \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Composition of Functions
Composition of functions involves combining two functions to create a new function. In this case, we have two functions: the volume V as a function of temperature s, and the temperature s as a function of time t. To find V as a function of t, we substitute the expression for s into the equation for V, effectively creating a new function that relates volume directly to time.
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Quadratic Functions
A quadratic function is a polynomial function of degree two, typically expressed in the form V = ax² + bx + c. In the given volume equation V = s² + 2s + 3, the variable s is raised to the second power, indicating that the volume changes in a parabolic manner with respect to temperature. Understanding the properties of quadratic functions, such as their vertex and axis of symmetry, is essential for analyzing their behavior.
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Substitution Method
The substitution method is a technique used in algebra and calculus to simplify expressions or solve equations by replacing a variable with another expression. In this problem, we substitute the expression for s (temperature) into the volume equation V. This method allows us to express V solely in terms of t (time), facilitating the analysis of how the balloon's volume changes over time.
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Related Practice
Textbook Question
Finding a Viewing Window
In Exercises 5–30, find an appropriate graphing software viewing window for the given function and use it to display that function’s graph. The window should give a picture of the overall behavior of the function. There is more than one choice, but incorrect choices can miss important aspects of the function.
y = 3 cos 60x
Textbook Question
Can a function be both even and odd? Give reasons for your answer.
Textbook Question
Graphing
In Exercises 69–76, graph each function not by plotting points, but by starting with the graph of one of the standard functions presented in Figures 1.14–1.17 and applying an appropriate transformation.
y = (−2x)²/³
1
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Textbook Question
Radians and Degrees
On a circle of radius 10 m, how long is an arc that subtends a central angle of (a) 4π/5 radians? (b) 110°?
Textbook Question
A hot-air balloon rising straight up from a level field is tracked by a range finder located 500 ft from the point of liftoff. Express the balloon’s height as a function of the angle the line from the range finder to the balloon makes with the ground.