Textbook Question
b. Slopes on a tangent curve What is the smallest value the slope of the curve can ever have on the interval −2 < x < 2? Give reasons for your answer.
Ch. 3 - Derivatives
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 3, Problem 3.2.36b
Average single-family home prices P (in thousands of dollars) in Sacramento, California, are shown in the accompanying figure from the beginning of 2006 through the end of 2015.
b. Estimate home prices at the end of
i) 2007 ii) 2012 iii) 2015
Verified step by step guidance1
To estimate the home prices at the end of 2007, locate the year 2007 on the horizontal axis (t-axis) of the graph. Follow the vertical line upwards until it intersects the curve. From this intersection point, move horizontally to the left to read the corresponding price on the vertical axis (P-axis).
For the year 2012, repeat the process: find 2012 on the t-axis, move vertically to the curve, and then horizontally to the P-axis to estimate the price.
Similarly, for the year 2015, locate 2015 on the t-axis, follow the vertical line to the curve, and then move horizontally to the P-axis to determine the estimated price.
Ensure that you are reading the values accurately by checking the scale of the axes. The vertical axis (P-axis) is marked in increments of 20, starting from 150 up to 390.
Note that these are estimates based on the graph, and the actual values may vary slightly depending on the precision of your reading from the graph.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Graph Interpretation
Understanding how to read and interpret graphs is crucial in calculus, especially when analyzing trends over time. In this case, the graph shows the average home prices in Sacramento from 2006 to 2015, highlighting key points such as peaks and troughs. Being able to extract numerical values from the graph at specific years is essential for estimating home prices.
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Graphing The Derivative
Estimation Techniques
Estimation techniques involve using visual data to approximate values that may not be explicitly stated. In this context, students need to estimate home prices at the end of 2007, 2012, and 2015 by analyzing the graph. This skill is important in calculus for making informed predictions based on available data.
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Trends and Patterns
Identifying trends and patterns in data is a fundamental concept in calculus and statistics. The graph illustrates a significant peak in home prices in 2007 followed by a decline until 2012, and a gradual recovery thereafter. Recognizing these trends helps in understanding the economic factors influencing home prices over time.
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06:21Properties of Functions
Related Practice
Textbook Question
Theory and Examples
In Exercises 51–54,
b. Graph y = f(x) and y = f'(x) side by side using separate sets of coordinate axes, and answer the following questions.
y = x⁴/4
Textbook Question
By computing the first few derivatives and looking for a pattern, find the following derivatives.
b. d¹¹⁰/dx¹¹⁰ (sin x − 3 cos x)
Textbook Question
Temperature The given graph shows the outside temperature T in °F, between 6 a.m. and 6 p.m.
b. At what time does the temperature increase most rapidly? Decrease most rapidly? What is the rate for each of those times?
Textbook Question
Common linear approximations at x = 0 Find the linearizations of the following functions at x = 0.
b. cos x
Textbook Question
Suppose that functions ƒ(x) and g(x) and their first derivatives have the following values at x = 0 and x = 1.
x ƒ(x) g(x) ƒ'(x) g'(x)
0 1 1 -3 1/2
1 3 5 1/2 -4
Find the first derivatives of the following combinations at the given value of x.
b. ƒ(x)g²(x), x = 0