Textbook Question
Let ƒ(x) = 3x - x³ . Show that the equation ƒ(𝓍) = -4 has a solution in the interval [2,3] and use Newton’s method to find it.
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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 68d
Theory and Examples
Sketch the graph of a differentiable function y = f(x) that has a local minima at (1, 1) and (3, 3).
Verified step by step guidance1
Start by understanding the concept of local minima. A local minimum is a point where the function value is lower than all nearby points. This means the derivative at these points is zero, indicating a horizontal tangent line.
Identify the given local minima points: (1, 1) and (3, 3). These points suggest that the function has a 'valley' or 'dip' at these coordinates.
Consider the behavior of the function around these points. Since the function is differentiable, it is smooth and continuous. The derivative changes sign around these minima points, from negative to positive.
Sketch the graph by plotting the points (1, 1) and (3, 3) on a coordinate plane. Draw a curve that dips down at these points, ensuring the tangent is horizontal at these minima.
Ensure the graph is smooth and continuous, reflecting the differentiability of the function. Between the minima, the graph should rise, and outside these points, it should fall, creating a 'U' shape around each minimum.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Differentiable Function
A differentiable function is one that has a derivative at every point in its domain. This means the function is smooth and continuous, without any sharp corners or cusps. Understanding differentiability is crucial for sketching graphs, as it ensures the function's behavior can be predicted using its derivative.
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05:53
Finding Differentials
Local Minima
A local minima of a function is a point where the function value is lower than all other nearby points. At a local minima, the derivative of the function changes from negative to positive, indicating a 'valley' in the graph. Identifying local minima is essential for sketching the graph accurately, as it highlights where the function reaches its lowest points locally.
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07:09The First Derivative Test: Finding Local Extrema
Graph Sketching
Graph sketching involves plotting the general shape of a function based on its critical points, such as local minima and maxima, and its behavior at infinity. It requires understanding the function's derivative to determine where the function is increasing or decreasing, and how it curves. This skill is vital for visualizing the function's behavior and ensuring the graph reflects all given conditions.
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11:41Summary of Curve Sketching
Related Practice
Textbook Question
The ladder problem What is the approximate length (in feet) of the longest ladder you can carry horizontally around the corner of the corridor shown here? Round your answer down to the nearest foot.
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Textbook Question
Particle motion The positions of two particles on the s-axis are s₁ = cos t and s₂ = cos (t + π/4) .
b. When do the particles collide?
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Textbook Question
Particle motion The positions of two particles on the s-axis are s₁ = cos t and s₂ = cos (t + π/4) .
a. What is the farthest apart the particles ever get?
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Textbook Question
Theory and Examples
Sketch the graph of a differentiable function y = f(x) that has a local maxima at (1, 1) and (3, 3)
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Textbook Question
Each of Exercises 89–92 shows the graphs of the first and second derivatives of a function y=f(x). Copy the picture and add to it a sketch of the approximate graph of f, given that the graph passes through the point P.
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