0
What else can I help you with?
What are the steps to solving an Optimization problem in calculus?
Optimization problems that involve finding the maxima or minima of functions also involve taking the first derivative of the function and finding locations where the value of the first derivative is equal to zero (by setting f'(x) = 0). The locations are either maxima, minima, or points of inflection and a second derivative test can be used to determine which of the three was found (max: f''(x) < 0, min: f''(x) > 0, PoI: f''(x) = 0).
Why there are two tests for checking maximum and minimum?
There are two tests involved in checking for maximum and minimum because, if you only checked for a value of zero for the first derivative, you would only know that the equation has zero slope at that point. You also need to check the second derivative to see if that point is a maximum, a minimum, or an inflection point.To be fully correct, you also need to understand the equation itself, because there may be more than one maxima or minima, and/or there may be a discontinuity. This is all part of the process of finding a maximum or minimum.
Is it always true that for any polynomial px if x is a zero of the derivative then x px is a maximum or minimum value of px?
No. The important decider is the second derivative of the polynomial (the gradient of the gradient of the polynomial) at the zero of the first derivative: If less than zero, then the point is a maximum If more than zero, then the point in a minimum If equal to zero, then the point is a point of inflection. Consider the polynomial f(x) = x3, then f'(x) = 3x2 f'(0) = 0 -> x = 0 could be a maximum, minimum or point of inflection. f''(x) = 6x f''(0) = 0 -> x = 0 is a point of inflection Points of inflection do not necessarily have a zero gradient, unlike maxima and minima which must. Points of inflection are the zeros of the second derivative of the polynomial.
What is the geometrical meaning for second derivative?
The Geometrical meaning of the second derivative is the curvature of the function. If the function has zero second derivative it is straight or flat.
How do you find second derivative of a function?
All it means to take the second derivative is to take the derivative of a function twice. For example, say you start with the function y=x2+2x The first derivative would be 2x+2 But when you take the derivative the first derivative you get the second derivative which would be 2
What is the geometrical meaning of second derivative?
The first derivative is the rate of change, and the second derivative is the rate of change of the rate of change.
What is the second derivative of a function's indefinite integral?
well, the second derivative is the derivative of the first derivative. so, the 2nd derivative of a function's indefinite integral is the derivative of the derivative of the function's indefinite integral. the derivative of a function's indefinite integral is the function, so the 2nd derivative of a function's indefinite integral is the derivative of the function.
What is the formula for calculating the parabola radius of curvature at a specific point on the curve?
The formula for calculating the parabola radius of curvature at a specific point on the curve is: R (1 (dy/dx)2)(3/2) / d2y/dx2, where R is the radius of curvature, dy/dx is the first derivative of y with respect to x, and d2y/dx2 is the second derivative of y with respect to x.
Why the second derivative negative for Maxima?
When you solve for the 2nd derivative, you are determining whether the function is concave up/down. If you calculated that the 2nd derivative is negative, the function is concave down, which means you have a relative/absolute maximum, given that the 1st derivative equals 0. To understand why this is, think about the definition of the 2nd derivative. It is a measure of the rate of change of the gradient. At a maximum, the gradient starts positive, becomes 0 at the maximum itself and then becomes negative, so it is decreasing. If the gradient is going down, then its rate of change, the 2nd derivative, must be negative.
What is the derivative of x to the second power?
2x is the first derivative of x2.
Does The second derivative represent the rate of change of the first derivative?
Yes.
What is the derivative of x to the second powerr?
2x is the first derivative of x2.
