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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 4 - Applications of DerivativesProblem 4.1.51a
Chapter 4, Problem 4.1.51a

Theory and Examples


In Exercises 51 and 52, give reasons for your answers.


Let f(x) = (x − 2)²ᐟ³.


a. Does f′(2) exist?

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To determine if f'(2) exists, we need to check if the function f(x) = (x - 2)^(2/3) is differentiable at x = 2. Differentiability at a point requires the function to be continuous at that point and the derivative to exist.
First, check the continuity of f(x) at x = 2. Since f(x) is defined for all real numbers and involves a power of (x - 2), it is continuous at x = 2.
Next, consider the definition of the derivative: f'(x) = lim (h -> 0) [(f(x + h) - f(x)) / h]. We need to evaluate this limit as x approaches 2.
Substitute f(x) = (x - 2)^(2/3) into the derivative definition: f'(2) = lim (h -> 0) [((2 + h - 2)^(2/3) - (2 - 2)^(2/3)) / h] = lim (h -> 0) [h^(2/3) / h].
Simplify the expression: h^(2/3) / h = h^(-1/3). As h approaches 0, h^(-1/3) becomes undefined because it involves division by zero. Therefore, the limit does not exist, and f'(2) does not exist.

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Key Concepts

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

Derivative

The derivative of a function at a point measures the rate at which the function's value changes as its input changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. For a derivative to exist at a point, the function must be continuous and smooth (differentiable) at that point.
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Derivatives

Continuity

Continuity of a function at a point means that the function is defined at that point, the limit of the function as it approaches the point from both sides exists, and the limit equals the function's value at that point. If a function is not continuous at a point, it cannot have a derivative there.
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Intro to Continuity

Cusp

A cusp is a point on the graph of a function where the direction of the curve changes abruptly, creating a sharp point. At a cusp, the function is continuous, but the derivative does not exist because the slope of the tangent line is not well-defined, as it approaches different values from the left and right.
Related Practice
Textbook Question

Identifying Extrema


In Exercises 41–52:


a. Identify the function’s local extreme values in the given domain, and say where they occur.


g(x) = −x² − 6x − 9,−4 ≤ x < ∞

Textbook Question

Identifying Extrema


In Exercises 53–60:


a. Find the local extrema of each function on the given interval, and say where they occur.


f(x) = sec²x − 2tan x, −π/2 < x < π/2

Textbook Question

Finding Antiderivatives

In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.

csc x cot x

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Textbook Question

Checking Antiderivative Formulas


Right, or wrong? Say which for each formula and give a brief reason for each answer.


∫tanθ sec²θ dθ = sec³θ / 3 + C

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Textbook Question

Identifying Extrema


In Exercises 63 and 64, the graph of f' is given. Assume that f has domain (-2, 2).


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a. Either use the graph to determine which intervals f is increasing on and which intervals f is decreasing on, or explain why this information cannot be determined from the graph.

Textbook Question

Finding Antiderivatives

In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.

(3/2)√x