- #1
PAllen said:compute dy/dx and plug into 2.10.b. It leads directly to 2.10. I certainly couldn't see it by eye - but it is a pretty small calculation.
Nano-Passion said:Sorry, I'm not able to read between the line here. What do you mean by compute dy/dx? Compute dy/dx of what?
PAllen said:You have x^2+y^2=R^2. Solve for y and take dy/dx. Plug this into where dy/dx is in 2.10.b. You can then simplify to 2.10.c.
Nano-Passion said:One more question, do you know how he got -1, 1 as the limits of integration? It looks as if he pulled that number randomly.
George Jones said:A change of integration variable was made such that the new integration variable is dimensionless, i.e., [itex]\xi = x/R[/itex].
Nano-Passion said:Okay, I also don't know how that integral will give you Pi. Something is missing in my knowledge-base.
PAllen said:That integral should be covered in any first course in calculus; or looked up in even the smallest table of integrals; or recognize that it is the circumference of a unit semi-circle. If you're reading this book, you should have calculus book, and can review integration of trigonometric forms.
Simple Derivation is a mathematical process of finding the derivative of a function. It is important because it allows us to calculate the instantaneous rate of change of a function at any given point, which has many practical applications in fields such as physics, engineering, and economics.
To solve a Simple Derivation problem, you need to follow a specific set of steps. First, identify the function and its independent variable. Then, use the power rule, product rule, quotient rule, or chain rule to find the derivative. Finally, simplify the derivative and substitute in the given value to find the instantaneous rate of change.
The most common mistakes in Simple Derivation include forgetting to apply the chain rule, using the wrong rule, not simplifying the derivative, and making mistakes in basic algebraic calculations. It is important to double-check your work and be careful with each step to avoid these mistakes.
To improve your understanding of Simple Derivation, it is important to practice solving various types of problems and to seek help when needed. You can also read textbooks or watch online tutorials to gain a better understanding of the concepts behind Simple Derivation.
Simple Derivation has many real-world applications, including determining the velocity and acceleration of an object, calculating the marginal cost and revenue in economics, and finding the rate of change of a chemical reaction in chemistry. It is also used in fields such as engineering, physics, and finance to model and analyze various systems and processes.