Textbook Question
Sketch the graphs of the rational functions in Exercises 53–60.
𝓍²
y = ------------------
𝓍² ― 4
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 4.3.33
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
Verified step by step guidance1
To find the intervals where the function is increasing or decreasing, first find the derivative of the function g(x) = x√(8 - x²). Use the product rule and chain rule to differentiate.
Set the derivative equal to zero to find the critical points. These points will help determine where the function changes from increasing to decreasing or vice versa.
Analyze the sign of the derivative on the intervals determined by the critical points. If the derivative is positive on an interval, the function is increasing there; if negative, the function is decreasing.
To identify local extrema, evaluate the function at the critical points. Use the first derivative test to determine if these points are local maxima or minima.
Summarize the intervals of increase and decrease, and list any local extrema along with their locations based on the analysis of the derivative and critical points.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative and Critical Points
The derivative of a function provides information about its rate of change. To find where a function is increasing or decreasing, calculate its derivative and identify critical points where the derivative is zero or undefined. These points are potential locations for local extrema and help determine intervals of increase or decrease.
Recommended video:
04:50
Critical Points
Increasing and Decreasing Intervals
A function is increasing on an interval if its derivative is positive over that interval, and decreasing if the derivative is negative. By analyzing the sign of the derivative across different intervals, you can determine where the function rises or falls, which is crucial for identifying behavior patterns in the function.
Recommended video:
07:32Determining Where a Function is Increasing & Decreasing
Local Extrema
Local extrema refer to the local maximum or minimum values of a function within a specific interval. These occur at critical points where the derivative changes sign. To identify local extrema, evaluate the function at critical points and use the first or second derivative test to confirm whether these points are maxima or minima.
Recommended video:
05:58Finding Extrema Graphically
Related Practice
Textbook Question
Finding Extrema from Graphs
In Exercises 11–14, match the table with a graph.
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.
∫7sin(θ/3) dθ
Textbook Question
Checking the Mean Value Theorem
Which of the functions in Exercises 7–12 satisfy the hypotheses of the Mean Value Theorem on the given interval, and which do not? Give reasons for your answers.
f(x) = {x² − x, −2 ≤ x ≤−1
2x² − 3x − 3, −1 < x ≤ 0
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Textbook Question
56. Airplane landing path An airplane is flying at altitude H when it begins its descent to an airport runway that is at horizontal ground distance L from the airplane, as shown in the accompanying figure. Assume that the landing path of the airplane is the graph of a cubic polynomial function y = ax^3+bx^2+cx+d, where y(-L)= H and y(0)=0.
a. What is dy/dx at x = 0?
b. What is dy/dx at x = -L?
c. Use the values for dy/dx at x = 0 and x =- L together with y(0) = 0 and y(-L) = H to show that y(x)=H[2(x/L)^3+3(x/L)^2]
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Textbook Question
Absolute Extrema on Finite Closed Intervals
In Exercises 21–36, find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.
g(x) = √(4 − x²), −2 ≤ x ≤ 1