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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 3.2.47c
Chapter 3, Problem 3.2.47c

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 open circles at x = -3 and x = 2, indicating points of discontinuity on the interval from -3 to 3.

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1
Examine the graph to identify any points where the function is not continuous. A function is not continuous at a point if there is a break, jump, or hole in the graph at that point.
Look for points where the function is not differentiable. A function is not differentiable at a point if there is a sharp corner or cusp, or if the function is not continuous at that point.
In the given graph, observe the open circles at x = -3 and x = 3, indicating that the function is not defined at these endpoints, hence not continuous.
Notice the sharp corner at x = 0, which suggests that the function is not differentiable at this point.
Conclude that the function is neither continuous nor differentiable at x = -3 and x = 3 due to the open circles, and not differentiable at x = 0 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. In simpler terms, there should be 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 means the function must be smooth and not have any sharp corners or vertical tangents. If a function is not continuous at a point, it cannot be differentiable there. Thus, differentiability implies continuity, but continuity does not necessarily imply differentiability.
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Finding Differentials

Points of Discontinuity

Points of discontinuity occur where a function is not continuous, which can happen due to removable discontinuities (holes), jump discontinuities, or infinite discontinuities (asymptotes). In the provided graph, the open circles at x = -3 and x = 2 indicate points where the function is not defined, suggesting that the function is neither continuous nor differentiable at these points.
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Critical Points
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

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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:


d. When does it speed up and slow down?


s = t³ - 6t² + 7t, 0 ≤ t ≤ 4

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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

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Textbook Question

Right circular cylinder The total surface area S of a right circular cylinder is related to the base radius r and height h by the equation S = 2πr² + 2πrh.


c. How is dS/dt related to dr/dt and dh/dt if neither r nor h is constant?

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Textbook Question

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


c. d⁷³/dx⁷³ (x sin x)

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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.

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