Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

2. Intro to Derivatives

Differentiability

Calculus

2. Intro to Derivatives

Differentiability

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1:24 minutes

Problem 3.2.45a

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

a. differentiable?

Give reasons for your answers.

Verified step by step guidance

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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Key Concepts

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

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.

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.

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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Related Practice

Textbook Question

One-Sided Derivatives

Compute the right-hand and left-hand derivatives as limits to show that the functions in Exercises 37–40 are not differentiable at the point P.

Textbook Question

One-Sided Derivatives

Compute the right-hand and left-hand derivatives as limits to show that the functions in Exercises 37–40 are not differentiable at the point P.

Textbook Question

In Exercises 41–44, determine whether the piecewise-defined function is differentiable at x = 0.

f(x) = { 2x − 1, x ≥ 0

x² + 2x + 7, x < 0

Textbook Question

In Exercises 41–44, determine whether the piecewise-defined function is differentiable at x = 0.

f(x) = { 2x + tan x, x ≥ 0

x², x < 0

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

a. differentiable?

Give reasons for your answers.

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

a. differentiable?

Give reasons for your answers.

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

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.

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Differentiability and Continuity on an IntervalEach 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 bea. differentiable?Give reasons for your answers.

Key Concepts

Differentiability

Continuity

Closed Interval

Watch next

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.