Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

5. Graphical Applications of Derivatives

Curve Sketching

Calculus

5. Graphical Applications of Derivatives

Curve Sketching

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3:10 minutes

Problem 4.4.78c

Textbook Question

{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.

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.

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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Textbook Question

24–34. Curve sketching Use the guidelines given in Section 4.4 to make a complete graph of the following functions on their domains or on the given interval. Use a graphing utility to check your work.

ƒ(x) = 10x² / (x² + 3)

Textbook Question

ƒ(x) = 4cos (π (x-1)) on [0, 2]

Textbook Question

{Use of Tech} ƒ(x) = x (x -1)e⁻ˣ

Textbook Question

{Use of Tech} A pursuit curve A man stands 1 mi east of a crossroads. At noon, a dog starts walking north from the crossroads at 1 mi/hr (see figure). At the same instant, the man starts walking and at all times walks directly toward the dog at s > 1 mi/hr . The path in the xy-plane followed by the man as he pursues the dog is given by the function y = ƒ(x) = s/2 ((x(ˢ⁺¹)/ˢ) /(s+1) - (x(ˢ⁺¹)/ˢ / s-1)) + s/ s² - 1

Select various values of s > 1 and graph this pursuit curve. Comment on the changes in the curve as s increases. <IMAGE>

Textbook Question

Graphs and Graphing

Graph the curves in Exercises 33–42.

y = 𝓍³ (8―𝓍 )

Textbook Question

Graphs and Graphing

Graph the curves in Exercises 33–42.

______

y = 𝓍√4 ― 𝓍²

Textbook Question

In Exercises 49–52, graph each function. Then use the function’s first derivative to explain what you see.