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Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblem 4.4.78c
Chapter 4, Problem 4.4.78c

{Use of Tech} Elliptic curves The equation y² = x³ - ax + 3, where a is a parameter, defines a well-known family of elliptic curves.


c. By experimentation, determine the approximate value of a (3 < a < 4)at which the graph separates into two curves.

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Understand the problem: We are given the equation of an elliptic curve y² = x³ - ax + 3, where 'a' is a parameter. We need to find the approximate value of 'a' between 3 and 4 where the graph separates into two distinct curves.
Recall that the separation of the graph into two curves occurs when the discriminant of the cubic polynomial in x, which is x³ - ax + 3, becomes zero. This is because the discriminant being zero indicates a double root, leading to a cusp or a singular point on the curve.
The discriminant Δ of a cubic polynomial x³ + bx + c is given by the formula Δ = -4b³ - 27c². In our case, b = -a and c = 3, so substitute these values into the discriminant formula.
Set the discriminant equal to zero to find the critical value of 'a': -4(-a)³ - 27(3)² = 0. Simplify this equation to solve for 'a'.
Experiment with values of 'a' between 3 and 4 to find the approximate value where the discriminant becomes zero, indicating the separation of the graph into two curves.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Elliptic Curves

Elliptic curves are defined by cubic equations in two variables, typically in the form y² = x³ + ax + b. They have important properties in number theory and algebraic geometry, particularly in relation to their group structure. The specific curve given, y² = x³ - ax + 3, is a family of elliptic curves parameterized by 'a', which affects the shape and intersection of the curve with the x-axis.
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Graph Behavior and Separation

The behavior of the graph of an elliptic curve can change based on the parameter 'a'. Specifically, the graph can separate into two distinct curves when the discriminant of the cubic equation changes sign, indicating a change in the number of real roots. This separation is crucial for understanding the nature of the solutions to the equation and can be explored through graphical or numerical methods.
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Numerical Experimentation

Numerical experimentation involves using computational tools or graphing techniques to explore mathematical phenomena. In this context, it means testing various values of 'a' within the specified range (3 < a < 4) to observe how the graph of the elliptic curve behaves. This approach helps in approximating the critical value of 'a' at which the curve separates, providing insights that may not be easily derived analytically.
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Related Practice
Textbook Question

107–110. {Use of Tech} Motion with gravity Consider the following descriptions of the vertical motion of an object subject only to the acceleration due to gravity. Begin with the acceleration equation a(t) = v' (t) = -g , where g = 9.8 m/s² .

d. Find the time when the object strikes the ground.

A payload is released at an elevation of 400 m from a hot-air balloon that is rising at a rate of 10 m/s.

Textbook Question

if ƒ(x) = 1 / (3x⁴ + 5) , it can be shown that ƒ'(x) = 12x³ / (3x⁴ + 5)² and ƒ"(x) = 180x² (x² + 1) (x + 1) (x - 1) / (3x⁴ + 5)³ . Use these functions to complete the following steps.


d. Identify the local extreme values and inflection points of ƒ .

Textbook Question

Population models The population of a species is given by the function P(t) = Kt²/(t² + b) , where t ≥ 0 is measured in years and K and b are positive real numbers.


c. For arbitrary positive values of K and b, when does the maximum growth rate occur (in terms of K and b)?

Textbook Question

107–110. {Use of Tech} Motion with gravity Consider the following descriptions of the vertical motion of an object subject only to the acceleration due to gravity. Begin with the acceleration equation a(t) = v' (t) = -g , where g = 9.8 m/s² .

c. Find the time when the object reaches its highest point. What is the height? 

A payload is released at an elevation of 400 m from a hot-air balloon that is rising at a rate of 10 m/s.

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Textbook Question

Rectangles in triangles Find the dimensions and area of the rectangle of maximum area that can be inscribed in the following figures.

d. An arbitrary triangle with a given area A (The result applies to any triangle, but first consider triangles for which all the angles are less than or equal to 90° .)

Textbook Question

{Use of Tech} Fixed points of quadratics and quartics Let f(x) = ax(1 -x), where a is a real number and 0 ≤ a ≤ 1. Recall that the fixed point of a function is a value of x such that f(x) = x (Exercises 48–51). 


c. Graph g for a = 2, 3, and 4.