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

4. Applications of Derivatives

Implicit Differentiation

Calculus

4. Applications of Derivatives

Implicit Differentiation

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3:04 minutes

Problem 70b

Textbook Question

The following equations implicitly define one or more functions.
b. Solve the given equation for y to identify the implicitly defined functions y=f₁(x), y = f₂(x), ….
y² = x²(4 − x) / 4 + x (right strophoid)

Verified step by step guidance

Start by isolating the term involving y on one side of the equation. The given equation is y² = x²(4 − x) / 4 + x. We need to solve for y, so we have y² = (x²(4 − x) / 4) + x.

To solve for y, take the square root of both sides of the equation. This gives us y = ±√((x²(4 − x) / 4) + x). Remember that taking the square root introduces both positive and negative solutions.

Now, express the solutions as functions of x. We have two functions: y = f₁(x) = √((x²(4 − x) / 4) + x) and y = f₂(x) = -√((x²(4 − x) / 4) + x).

Consider the domain of the functions. Since we are taking the square root, the expression inside the square root must be non-negative. Set up the inequality (x²(4 − x) / 4) + x ≥ 0 and solve for x to find the domain.

Verify the solutions by substituting back into the original equation to ensure they satisfy y² = x²(4 − x) / 4 + x. This step confirms that the functions y = f₁(x) and y = f₂(x) are correctly derived from the implicit equation.

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Key Concepts

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

Implicit Functions

Implicit functions are defined by equations where the dependent variable is not isolated on one side. In the context of calculus, these functions can often be expressed in terms of the independent variable through algebraic manipulation. Understanding how to manipulate these equations is crucial for identifying the functions they define.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations to isolate variables. This skill is essential when solving for y in an implicit equation, as it allows one to express y explicitly in terms of x. Techniques such as factoring, expanding, and using the quadratic formula may be necessary to achieve this.

Quadratic Equations

Quadratic equations are polynomial equations of the form ax² + bx + c = 0, where a, b, and c are constants. In the context of the given equation, recognizing that it can be rearranged into a quadratic form in terms of y is vital. Solutions to quadratic equations can yield multiple values for y, representing different branches of the implicitly defined function.