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Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 3 - DerivativesProblema 3.2.45a
Capitolo 3, Problema 3.2.45a

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


a. differentiable?


Give reasons for your answers.
Graph of a linear function with endpoints at (-3, 2) and (2, -2) over the interval from -3 to 2 on a coordinate plane.

Guida verificata passo dopo passo
1
Examine the graph of the function over the interval D: [-3, 2]. The function is represented by a straight line, which suggests it is a linear function.
Recall that a function is differentiable at a point if it is continuous at that point and has a defined derivative there. Linear functions are differentiable everywhere on their domain.
Check the endpoints of the interval, x = -3 and x = 2. At x = -3, the function is continuous and the derivative exists because the line extends smoothly from this point. At x = 2, the function is also continuous and the derivative exists as the line ends smoothly.
Since the function is a straight line, it is differentiable at every point in the interval D: [-3, 2]. There are no corners, cusps, or vertical tangents that would make the function non-differentiable at any point.
Conclude that the function is differentiable at all points in the domain D: [-3, 2] because it is a linear function, which is inherently differentiable across its entire domain.

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Differentiability

A function is differentiable at a point if it has a defined derivative at that point, meaning the function's graph has a tangent line that is well-defined. This requires the function to be continuous at that point and for the left-hand and right-hand derivatives to be equal. Points where the graph has sharp corners, vertical tangents, or discontinuities indicate non-differentiability.
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Finding Differentials

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

Closed Interval

A closed interval, denoted as [a, b], includes all numbers between a and b, as well as the endpoints a and b themselves. In the context of calculus, analyzing functions over closed intervals is important for determining properties like continuity and differentiability at the endpoints, which can behave differently than points within the interval.
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Improper Integrals: Infinite Intervals
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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


a. differentiable?


Give reasons for your answers.

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The radius r of a circle is measured with an error of at most 2%. What is the maximum corresponding percentage error in computing the circle’s


a. circumference?

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Find y'' if:


a. y = csc x

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Quadratic approximations


a. Let Q(x) = b₀ + b₁(x − a) + b₂(x − a)² be a quadratic approximation to f(x) at x = a with these properties:


i. Q(a) = f(a)

ii. Q′(a) = f′(a)

iii. Q″(a) = f″(a).


Determine the coefficients b₀, b₁, and b₂.

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Interpreting Derivative Values


Effectiveness of a drug On a scale from 0 to 1, the effectiveness E of a pain-killing drug t hours after entering the bloodstream is displayed in the accompanying figure.


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a. At what times does the effectiveness appear to be increasing? What is true about the derivative at those times?

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If the original 24 m edge length x of a cube decreases at the rate of 5 m/min, when x = 3 m at what rate does the cube’s

a. surface area change?

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