Ch. 4 - Applications of Derivatives

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 4 - Applications of Derivatives

Problem 4.10

Chapter 4, Problem 4.10

In Exercises 1–10, find the extreme values (absolute and local) of the function over its natural domain, and where they occur.

y = (𝓍 + 1) / (𝓍² + 2𝓍 + 2)

Verified step by step guidance

First, identify the natural domain of the function y = (𝓍 + 1) / (𝓍² + 2𝓍 + 2). The denominator must not be zero, so solve 𝓍² + 2𝓍 + 2 = 0 to find any restrictions on 𝓍.

Next, find the derivative of the function y with respect to 𝓍 to determine critical points. Use the quotient rule: if y = u/v, then y' = (u'v - uv') / v², where u = 𝓍 + 1 and v = 𝓍² + 2𝓍 + 2.

Set the derivative equal to zero to find critical points. Solve the equation (u'v - uv') = 0 for 𝓍, which will give you the values of 𝓍 where the slope of the tangent is zero, indicating potential extreme values.

Evaluate the second derivative to determine the concavity at the critical points. If the second derivative is positive at a critical point, the function has a local minimum there; if negative, a local maximum.

Finally, analyze the behavior of the function as 𝓍 approaches the boundaries of the domain or infinity to determine if there are any absolute extreme values. Consider the limits of y as 𝓍 approaches any boundary values or infinity.

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Key Concepts

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

Critical Points

Critical points of a function occur where its derivative is zero or undefined. These points are potential candidates for local extrema. To find them, differentiate the function with respect to x, set the derivative equal to zero, and solve for x. Additionally, check where the derivative does not exist, as these points may also indicate critical points.

First Derivative Test

The First Derivative Test helps determine whether a critical point is a local maximum, minimum, or neither. By analyzing the sign of the derivative before and after the critical point, one can infer the behavior of the function. If the derivative changes from positive to negative, the point is a local maximum; if it changes from negative to positive, it's a local minimum.

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For rational functions, like the one given, the domain excludes values that make the denominator zero. Understanding the domain is crucial for identifying where to look for extrema, as it defines the interval over which the function is analyzed.

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Related Practice

Textbook Question

In Exercises 1–10, find the extreme values (absolute and local) of the function over its natural domain, and where they occur.

__________

y = √ 3 + 2𝓍 ―𝓍²

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Textbook Question

Each of Exercises 67–88 gives the first derivative of a continuous function y=f(x). Find y'' and then use Steps 2–4 of the graphing procedure described in this section to sketch the general shape of the graph of f.

69. y' = x(x - 3)²

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Textbook Question

6. You are planning to close off a corner of the first quadrant with a line segment 20 units long running from (a, 0) to (0,b). Show that the area of the triangle enclosed by the segment is largest when a = b.

213

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Textbook Question

Identifying Extrema

In Exercises 19–40:

a. Find the open intervals on which the function is increasing and those on which it is decreasing.

b. Identify the function’s local extreme values, if any, saying where they occur.

g(x) = x√8 − x²

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Textbook Question

Finding Indefinite Integrals

In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫(√x + ³√x) dx

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Textbook Question

Finding Indefinite Integrals

In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫(t√t + √t) / t² dt

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In Exercises 1–10, find the extreme values (absolute and local) of the function over its natural domain, and where they occur.y = (𝓍 + 1) / (𝓍² + 2𝓍 + 2)

Verified video answer for a similar problem:

Key Concepts

Critical Points

First Derivative Test

Domain of a Function

In Exercises 1–10, find the extreme values (absolute and local) of the function over its natural domain, and where they occur.

y = (𝓍 + 1) / (𝓍² + 2𝓍 + 2)