Ch. 4 - Applications of Derivatives

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

Hass 15th Edition

Ch. 4 - Applications of Derivatives

Problem 4.3.64a

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 guidance

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).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

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).

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.

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:

06:15

Graphing The Derivative

Related Practice

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

Theory and Examples

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

Let f(x) = (x − 2)²ᐟ³.

a. Does f′(2) exist?

Textbook Question

Theory and Examples

Sketch the graph of a differentiable function y = f(x) that has a local minimum at (1, 1) and a local maximum at (3, 3).

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

views

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

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.

f(x) = √(x² − 2x − 3), 3 ≤ x < ∞

Identifying ExtremaIn 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 video answer for a similar problem:

Key Concepts

First Derivative Test

Critical Points

Graph Interpretation

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.