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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 3.68
Chapter 3, Problem 3.68

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
1
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.
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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).
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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.
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Related Practice
Textbook Question

In Exercises 41–58, find dy/dt.


y = (1 + tan⁴(t/12))³

1
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Textbook Question

Find dr/dθ in Exercises 15–18.


r – 2√θ = (3/2)θ²/³ + (4/3)θ³/⁴

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Derivative Calculations


In Exercises 1–8, given y = f(u) and u = g(x), find dy/dx = f'(g(x)) g'(x).


y = √u, u = sin x

Textbook Question

For Exercises 55 and 56, evaluate each limit by first converting each to a derivative at a particular x-value.


lim (x → 1) (x⁵⁰ − 1) / (x − 1)

Textbook Question

Derivatives in Differential Form


In Exercises 17–28, find dy.


y = sec(x² − 1)

Textbook Question

Is there a value of c that will make


f(x) = { (sin²(3x)) / x², x ≠ 0

c, x = 0


continuous at x = 0? Give reasons for your answer.