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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 4.3.1a
Chapter 4, Problem 4.3.1a

Analyzing Functions from Derivatives


Answer the following questions about the functions whose derivatives are given in Exercises 1–14:


a. What are the critical points of f?


f′(x) = x(x − 1)

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1
To find the critical points of a function, we need to determine where its derivative is equal to zero or undefined. In this case, the derivative is given as f'(x) = x(x - 1).
Set the derivative equal to zero to find the critical points: x(x - 1) = 0.
Solve the equation x(x - 1) = 0 by setting each factor equal to zero: x = 0 and x - 1 = 0.
Solving these equations gives the critical points: x = 0 and x = 1.
Since the derivative is a polynomial, it is defined for all real numbers, so there are no points where the derivative is undefined. Therefore, the critical points are x = 0 and x = 1.

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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 significant because they can indicate local maxima, minima, or points of inflection. To find critical points, set the derivative equal to zero and solve for the variable, or identify where the derivative does not exist.
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Critical Points

Derivative

The derivative of a function represents the rate of change or the slope of the function at any given point. It is a fundamental tool in calculus used to find critical points, analyze function behavior, and solve optimization problems. In this context, f′(x) = x(x − 1) is the derivative of the function f(x).
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Derivatives

Factoring

Factoring is a mathematical process used to simplify expressions or solve equations by expressing a polynomial as a product of its factors. In the context of finding critical points, factoring the derivative, such as f′(x) = x(x − 1), helps identify the values of x that make the derivative zero, which are potential critical points.
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Limits of Rational Functions: Denominator = 0
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.


k(x) = x³ + 3x² + 3x + 1, −∞ < x ≤ 0

Textbook Question

Finding displacement from an antiderivative of velocity


a. Suppose that the velocity of a body moving along the s-axis is


ds/dt = v = 9.8t − 3.


i. Find the body’s displacement over the time interval from t = 1 to t = 3 given that s = 5 when t = 0.

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.

−πsin πx

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.

sec²x

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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) = sin x − cos x,0 ≤ x ≤ 2π

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.


f(r) = 3r³ + 16r