Textbook Question
Initial Value Problems
Solve the initial value problems in Exercises 71–90.
d²r/dt² = 2/t³; dr/dt|ₜ ₌ ₁ =1, r(1) = 1
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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 4.1.65
Theory and Examples
[Technology Exercise] Graph the functions in Exercises 63–66. Then find the extreme values of the function on the interval and say where they occur.
h(x) = |x + 2| − |x − 3|, −∞ < x < ∞
Verified step by step guidance1
Step 1: Understand the function h(x) = |x + 2| − |x − 3|. This function involves absolute values, which can be split into piecewise functions based on the critical points where the expressions inside the absolute values change sign.
Step 2: Identify the critical points for the absolute value expressions. The critical points are x = -2 and x = 3, where the expressions inside the absolute values change sign.
Step 3: Break down the function into piecewise components based on the critical points. For x < -2, both expressions are negative, for -2 ≤ x < 3, the first expression is non-negative and the second is negative, and for x ≥ 3, both expressions are non-negative.
Step 4: Graph the piecewise function by evaluating h(x) in each interval: x < -2, -2 ≤ x < 3, and x ≥ 3. This will help visualize the behavior of the function across the entire domain.
Step 5: Determine the extreme values by analyzing the graph and evaluating the function at the critical points and endpoints of the intervals. The extreme values occur where the function reaches its maximum or minimum values within the given domain.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Absolute Value Functions
Absolute value functions, such as |x + 2| and |x - 3|, measure the distance of a number from zero on the number line, always yielding a non-negative result. These functions create V-shaped graphs and are crucial for understanding how the function h(x) = |x + 2| − |x − 3| behaves, especially at points where the expressions inside the absolute values change sign.
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06:37
Average Value of a Function
Graphing Piecewise Functions
Graphing piecewise functions involves plotting different expressions over specified intervals. For h(x) = |x + 2| − |x − 3|, the function can be broken into segments based on the critical points where x + 2 = 0 and x - 3 = 0. Understanding how to graph these segments helps visualize the function's behavior and identify extreme values.
Recommended video:
05:36Piecewise Functions
Finding Extreme Values
Extreme values of a function are its maximum and minimum values within a given interval. To find these for h(x), analyze the critical points where the derivative is zero or undefined, and evaluate the function at these points and endpoints of the interval. This process helps determine where the function reaches its highest or lowest values.
Recommended video:
05:12Finding Global Extrema (Extreme Value Theorem)
Related Practice
Textbook Question
Identifying Extrema
In Exercises 15–18:
a. Find the open intervals on which the function is increasing and those on which it is decreasing.
b. Identify the function’s local and absolute extreme values, if any, saying where they occur.
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Textbook Question
Theory and Examples
[Technology Exercise] Graph the functions in Exercises 63–66. Then find the extreme values of the function on the interval and say where they occur.
f(x) = |x − 2| + |x + 3|, −5 ≤ x ≤ 5
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Textbook Question
Each of Exercises 67–88 gives the first derivative of a continuous function y=f(x). Find y'' and then use Steps 2–4 of the graphing procedure described in this section to sketch the general shape of the graph of f.
74. y' = (x² - 2x)(x - 5)²
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Textbook Question
Finding Extreme Values
In Exercises 1–10, find the extreme values (absolute and local) of the function over its natural domain, and where they occur.
y = 𝓍³ + 𝓍² ― 8𝓍 + 5
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Textbook Question
Identify the inflection points and local maxima and minima of the functions graphed in Exercises 1–8. Identify the open intervals on which the functions are differentiable and the graphs are concave up and concave down.
4. y=9/14x^(1/3)(x^2-7)
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