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Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 3 - DerivativesProblem 3.2.48c
Chapter 3, Problem 3.2.48c

Differentiability and Continuity on an Interval


Each figure in Exercises 45–50 shows the graph of a function over a closed interval D. At what domain points does the function appear to be


c. neither continuous nor differentiable?


Give reasons for your answers.


Graph of a function with marked points indicating continuity and differentiability over the interval from -2 to 3.

Verified step by step guidance
1
Examine the graph for any points where there is a jump, hole, or vertical asymptote, as these indicate discontinuity.
Identify the point at x = 0, where there is a hole in the graph, indicating the function is not continuous at this point.
Check for differentiability by looking for sharp corners or cusps in the graph, as these indicate non-differentiability.
Notice the sharp corner at x = -1, which suggests the function is not differentiable at this point.
Conclude that the function is neither continuous nor differentiable at x = 0 due to the hole, and not differentiable at x = -1 due to the sharp corner.

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Key Concepts

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

Continuity

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. This means there are no breaks, jumps, or holes in the graph at that point. For a function to be continuous over an interval, it must be continuous at every point within that interval.
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Intro to Continuity

Differentiability

A function is differentiable at a point if it has a defined derivative at that point, which implies that the function must be continuous there. Differentiability also requires that the graph has a tangent line at that point, meaning it cannot have sharp corners or vertical tangents. If a function is not continuous at a point, it cannot be differentiable there.
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Finding Differentials

Points of Non-Continuity and Non-Differentiability

Points where a function is neither continuous nor differentiable typically include locations where there are jumps, vertical tangents, or corners in the graph. In the provided graph, such points can be identified by observing where the function does not meet the criteria for continuity or where the slope abruptly changes, indicating a lack of differentiability.
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Intro to Continuity Example 1
Related Practice
Textbook Question

Motion Along a Coordinate Line


Exercises 1–6 give the positions s = f(t) of a body moving on a coordinate line, with s in meters and t in seconds.


c. When, if ever, during the interval does the body change direction?


s = 25/(t + 5), −4 ≤ t ≤ 0

Textbook Question

Analyzing Motion Using Graphs


[Technology Exercise] Exercises 31–34 give the position function s = f(t) of an object moving along the s-axis as a function of time t. Graph f together with the velocity function v(t) = ds/dt = f'(t) and the acceleration function a(t) = d²s/dt² = f''(t). Comment on the object’s behavior in relation to the signs and values of v and a. Include in your commentary such topics as the following:


c. When does it change direction?


s = t² - 3t + 2, 0 ≤ t ≤ 5

Textbook Question

Suppose that the functions f and g and their derivatives with respect to x have the following values at x = 0 and x = 1.


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Find the derivatives with respect to x of the following combinations at the given value of x.


c. f(x) / (g(x) + 1), x = 1

Textbook Question

Differentiability and Continuity on an Interval


Each figure in Exercises 45–50 shows the graph of a function over a closed interval D. At what domain points does the function appear to be


c. neither continuous nor differentiable?


Give reasons for your answers.


Textbook Question

By computing the first few derivatives and looking for a pattern, find the following derivatives.


c. d⁷³/dx⁷³ (x sin x)

Textbook Question

Particle motion At time t, the position of a body moving along the s-axis is s = t³ − 6t² + 9t m.


c. Find the total distance traveled by the body from t = 0 to t = 2.