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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 4 - Applications of DerivativesProblema 4.1.50
Capitolo 4, Problema 4.1.50

Finding Critical Points


In Exercises 41–50, determine all critical points and all domain endpoints for each function.


y = x − 3x²ᐟ³

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Step 1: Understand the concept of critical points. Critical points occur where the derivative of a function is zero or undefined. These points are potential locations for local maxima, minima, or points of inflection.
Step 2: Find the derivative of the function y = x - 3x^(2/3). Use the power rule and the constant multiple rule to differentiate each term separately.
Step 3: Set the derivative equal to zero to find where the slope of the tangent is horizontal. Solve the equation for x to find potential critical points.
Step 4: Determine where the derivative is undefined. This occurs when the denominator of any fraction in the derivative is zero or when the derivative involves a root that results in an undefined expression.
Step 5: Identify the domain of the original function y = x - 3x^(2/3). Consider any restrictions on x that might affect the existence of critical points, such as values that make the function undefined.

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Critical Points

Critical points of a function occur where its derivative is zero or undefined. These points are essential for identifying local maxima, minima, or points of inflection. To find critical points, compute the derivative of the function and solve for values of x where the derivative equals zero or does not exist.
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Critical Points

Derivative

The derivative of a function represents the rate of change of the function with respect to its variable. It is a fundamental tool in calculus used to find slopes of tangent lines, velocities, and optimize functions. For the function y = x − 3x²ᐟ³, the derivative helps identify critical points by setting it to zero or finding where it is undefined.
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Derivatives

Domain Endpoints

Domain endpoints are the boundary values of the domain of a function, where the function is defined. These endpoints are crucial when analyzing the behavior of a function over its entire domain, especially when determining absolute extrema. For y = x − 3x²ᐟ³, consider the domain of x and identify any endpoints that might affect the function's critical points.
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Finding the Domain and Range of a Graph
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[Technology Exercises] When solving Exercises 14–30, you may need to use appropriate technology (such as a calculator or a computer).

26. Factoring a quartic Find the approximate values of r_1 through r_4 in the factorization

8x^4-14x^3-9x^2+11x-1=8(x-r_1)(x-r_2)(x-r_3)(x-r_4)

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Roots (Zeros)


Show that the functions in Exercises 19–26 have exactly one zero in the given interval.


f(x) = x³ + 4x² + 7, (−∞, 0)

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Checking Antiderivative Formulas


Verify the formulas in Exercises 57–62 by differentiation.


∫(3x + 5)⁻² dx = −(3x + 5)⁻¹/3 + C

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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.


∫csc θ/(csc θ − sin θ) dθ

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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.


∫(−3csc²x)dx

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Initial Value Problems

Solve the initial value problems in Exercises 71–90.

d³y/dx³ = 6; y″(0) = −8, y′(0) = 0, y(0) = 5

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