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
28
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Ch. 4 - Applications of Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 4, Problem 4.3.64a
Identifying Extrema
In Exercises 63 and 64, the graph of f' is given. Assume that f has domain (-2, 2).
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.
Verified step by step guidance1
Step 1: Recall that the graph provided is of f'(x), the derivative of f(x). The derivative indicates the slope of the tangent line to the graph of f(x). When f'(x) > 0, f(x) is increasing, and when f'(x) < 0, f(x) is decreasing.
Step 2: Analyze the graph of f'(x). Identify the intervals where f'(x) is positive (above the x-axis) and where f'(x) is negative (below the x-axis). This will help determine the intervals of increase and decrease for f(x).
Step 3: From the graph, observe that f'(x) is positive on the interval (-2, -1) and (0, 1). This means f(x) is increasing on these intervals.
Step 4: Similarly, observe that f'(x) is negative on the interval (-1, 0) and (1, 2). This means f(x) is decreasing on these intervals.
Step 5: Summarize the findings: f(x) is increasing on (-2, -1) and (0, 1), and f(x) is decreasing on (-1, 0) and (1, 2). This information is determined directly from the graph of f'(x).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
First Derivative Test
The First Derivative Test is used to determine where a function is increasing or decreasing. If the derivative f'(x) is positive over an interval, the function f(x) is increasing on that interval. Conversely, if f'(x) is negative, f(x) is decreasing. This test helps identify local maxima and minima by analyzing the sign changes of f'(x).
Recommended video:
07:09
The First Derivative Test: Finding Local Extrema
Critical Points
Critical points occur where the derivative f'(x) is zero or undefined. These points are potential locations for local extrema (maxima or minima) of the function f(x). By examining the behavior of f'(x) around these points, one can determine the nature of the extrema using the First Derivative Test or the Second Derivative Test.
Recommended video:
04:50Critical Points
Graph Interpretation
Interpreting the graph of a derivative involves understanding how the slope of the tangent line to the function f(x) changes. The graph of f'(x) provides visual cues about the intervals of increase and decrease of f(x). Positive values of f'(x) indicate increasing intervals, while negative values indicate decreasing intervals, helping to identify the behavior of the original function.
Recommended video:
06:15Graphing The Derivative
Related Practice
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
15
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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) = (x − 2) / (x²−1), 0 ≤ x < 1
208
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Textbook Question
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) = 1− 4/x², x ≠ 0
134
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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
29
views
Textbook Question
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)
149
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