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Ch. 2 - Limits
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 2 - LimitsProblem 2.7.3
Chapter 2, Problem 2.7.3

Which one of the following intervals is not symmetric about x=5?
a.(1, 9)
b.(4, 6)
c.(3, 8)
d.(4.5, 5.5)

Verified step by step guidance
1
Step 1: Understand the concept of symmetry about a point. An interval is symmetric about a point x = a if the distance from a to the left endpoint is equal to the distance from a to the right endpoint.
Step 2: For each interval, calculate the midpoint and check if it is equal to x = 5.
Step 3: For interval (1, 9), calculate the midpoint: (1 + 9) / 2.
Step 4: For interval (4, 6), calculate the midpoint: (4 + 6) / 2.
Step 5: For interval (3, 8), calculate the midpoint: (3 + 8) / 2.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Symmetry in Intervals

An interval is symmetric about a point if the distances from that point to the endpoints of the interval are equal. For an interval (a, b) to be symmetric about x = c, it must hold that c - a = b - c. This means that the midpoint of the interval must coincide with the point of symmetry.
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Endpoints of an Interval

The endpoints of an interval are the values that define the limits of the interval. For example, in the interval (a, b), 'a' is the left endpoint and 'b' is the right endpoint. Understanding the endpoints is crucial for determining the symmetry of the interval about a specific point.
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Calculating Midpoints

The midpoint of an interval (a, b) can be calculated using the formula (a + b) / 2. This value represents the center of the interval and is essential for assessing symmetry. If the midpoint does not equal the point of symmetry, the interval is not symmetric about that point.
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Related Practice
Textbook Question

Determine the end behavior of the following transcendental functions by analyzing appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist. 

f(x)=sinxf\(\left\)(x\(\right\))=\(\sin\) x

Textbook Question

Evaluate each limit. 


limx→3x2+7{\(\displaystyle\]\lim\)_{x\(\to\)3}\(\sqrt{x^2+7}{}\)}

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Textbook Question

Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.

lim x→3 x − 3 /|x − 3|

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Textbook Question

Evaluate each limit and justify your answer. 

lim x→2 (3 / 2x^5−4x^2−50)^4

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Textbook Question

Use the definitions given in Exercise 57 to prove the following infinite limits.


lim x→1^+ 1 /1 − x=−∞

Textbook Question

Determine the following limits.


limx→0x3−5x2x2{\(\displaystyle\]\lim\)_{x\(\to\)0}}\(\frac{x^3-5x^2}{x^2}\)