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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2:06 minutes

Problem 3.68

Textbook Question

a. Graph the function

ƒ(x) = { x, -1 ≤ x < 0
{ tan x, 0 ≤ x ≤ π/4.

b. Is ƒ continuous at x = 0?
c. Is ƒ differentiable at x = 0?

Give reasons for your answers.

Verified step by step guidance

Step 1: To graph the function ƒ(x), first consider the piecewise definition. For the interval -1 ≤ x < 0, the function is ƒ(x) = x, which is a linear function. Plot this part of the graph as a straight line starting at x = -1 and ending just before x = 0.

Step 2: For the interval 0 ≤ x ≤ π/4, the function is ƒ(x) = tan(x). Plot this part of the graph starting at x = 0 and ending at x = π/4. Note that tan(x) is continuous and differentiable within this interval.

Step 3: To determine if ƒ is continuous at x = 0, check the left-hand limit and right-hand limit as x approaches 0. The left-hand limit is the value of ƒ(x) as x approaches 0 from the left, which is 0. The right-hand limit is the value of ƒ(x) as x approaches 0 from the right, which is tan(0) = 0. Since both limits are equal and ƒ(0) = tan(0) = 0, ƒ is continuous at x = 0.

Step 4: To determine if ƒ is differentiable at x = 0, check the derivative from the left and right. The derivative of ƒ(x) = x for x < 0 is 1, and the derivative of ƒ(x) = tan(x) for x > 0 is sec²(x). Evaluate sec²(x) at x = 0, which is 1. Since both derivatives are equal at x = 0, ƒ is differentiable at x = 0.

Step 5: Summarize the findings: The function ƒ(x) is continuous at x = 0 because the left-hand limit, right-hand limit, and ƒ(0) are all equal. It is differentiable at x = 0 because the derivatives from the left and right are equal at that point.

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Key Concepts

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

Piecewise Functions

A piecewise function is defined by different expressions based on the input value. In this case, the function ƒ(x) has two distinct parts: one for values of x from -1 to 0, and another for values from 0 to π/4. Understanding how to evaluate and graph each piece is crucial for analyzing the overall behavior of the function.

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. For ƒ(x) to be continuous at x = 0, we need to check if the left-hand limit (as x approaches 0 from the left) equals the right-hand limit (as x approaches 0 from the right) and if both equal ƒ(0).

Differentiability

A function is differentiable at a point if it has a defined derivative at that point, which requires the function to be continuous there. To determine if ƒ(x) is differentiable at x = 0, we must examine the left-hand and right-hand derivatives at that point and see if they are equal. If they differ, the function is not differentiable at that point.