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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 4.4.78a
Chapter 4, Problem 4.4.78a

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


a. Plot a graph of the curve when a = 3.

Verified step by step guidance
1
Understand the equation of the elliptic curve: The given equation is y² = x³ - ax + 3, where 'a' is a parameter. For this problem, we need to consider the case when a = 3.
Substitute the value of 'a' into the equation: Replace 'a' with 3 in the equation to get y² = x³ - 3x + 3.
Choose a range of x-values: To plot the graph, select a range of x-values. A common choice might be from -5 to 5, but you can adjust this range based on the desired detail of the graph.
Calculate corresponding y-values: For each x-value in your chosen range, calculate the corresponding y-values using the equation y² = x³ - 3x + 3. Remember that y can be positive or negative since y² is involved.
Plot the points and sketch the curve: Using the calculated (x, y) pairs, plot these points on a graph. Connect the points smoothly to visualize the elliptic curve. Ensure to consider both positive and negative y-values for each x to capture the full curve.

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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 smooth, projective algebraic curves of genus one, equipped with a specified point at infinity. They are defined by equations of the form y² = x³ + ax + b, where the coefficients a and b satisfy certain conditions to ensure the curve has no singular points. These curves have important applications in number theory, cryptography, and complex analysis.
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Summary of Curve Sketching

Graphing Functions

Graphing functions involves plotting points on a coordinate plane to visualize the relationship between variables. For the elliptic curve defined by y² = x³ - ax + 3, one must compute y for various x values, taking care to consider both positive and negative roots of y². This process helps in understanding the shape and properties of the curve.
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Graph of Sine and Cosine Function

Parameter Variation

Parameter variation refers to how changing a parameter in an equation affects the graph of the function. In the case of the elliptic curve y² = x³ - ax + 3, varying the parameter 'a' alters the curve's shape and position. Analyzing these changes is crucial for understanding the family of curves defined by different values of 'a'.
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Critical Points Example 2
Related Practice
Textbook Question

{Use of Tech} A damped oscillator The displacement of an object as it bounces vertically up and down on a spring is given by y(t) = 2.5e⁻ᵗ cos 2t, where the initial displacement is y(0) = 2.5 and y = 0 corresponds to the rest position (see figure). <IMAGE>

a. Find the time at which the object first passes the rest position, y = 0. 

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Textbook Question

Rectangles beneath a line


a. A rectangle is constructed with one side on the positive x-axis, one side on the positive y-axis, and the vertex opposite the origin on the line y = 10 - 2x. What dimensions maximize the area of the rectangle? What is the maximum area?

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Textbook Question

Pen problems


b. A rancher plans to make four identical and adjacent rectangular pens against a barn, each with an area of 100 m² (see figure). What are the dimensions of each pen that minimize the amount of fence that must be used? <IMAGE>

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Textbook Question

Sketch a graph of a function f with the following properties.


f' < 0 and f" < 0, for x < -1

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Textbook Question

{Use of Tech} Every second counts You must get from a point P on the straight shore of a lake to a stranded swimmer who is 50 from a point Q on the shore that is 50 m from you (see figure). Assuming that you can swim at a speed of 2 m/s and run at a speed of 4 m/s, the goal of this exercise is to determine the point along the shore, x meters from Q, where you should stop running and start swimming to reach the swimmer in the minimum time. <IMAGE>


a. Find the function T that gives the travel time as a function of x, where 0 ≤ x ≤ 50.

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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 .

a. Find the velocity of the object for all relevant times. 

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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