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How can you know that a value is the minimum of a function?
When the first derivative of the function is equal to zero and the second derivative is positive.
What are the uses of derivatives?
A derivative of a function represents that equation's slope at any given point on its graph.
How do you find points of inflection in calculus?
Points of inflection on curves are where the curvature changes sign, such as when the second deriviative changes sign
Why the derivative is set equal to zero?
The first derivative is set to zero to find the critical points of the function. A critical point can be a minimum, maximum, or a saddle point. There's a reason for this. Suppose a differentiable function f:R->R has a maximum at x=a. Then the function goes down to the right of a, which means f'(a)
What is the derivative of ln cos x?
The derivative of the natural log is 1/x, therefore the derivative is 1/cos(x). However, since the value of cos(x) is submitted within the natural log we must use the chain rule. Then, we multiply 1/cos(x) by the derivative of cos(x). We get the answer: -sin(x)/cos(x) which can be simplified into -tan(x).
Why do you need to find the inflection point on a graph?
To find the inflection points on a graph, you need to take the second derivative. Then, set that equal to zero to find the x value(s) of the inflection point(s).
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 to do when second derivative is equal to zero while calculating Maxima and Minima?
Plot the function. You may have found an inflection point.
Is inflection point the saddle point?
An inflection point is not a saddle point, but a saddle point is an inflection point. To be precise, a saddle point is both a stationary point and an inflection point. An inflection point is a point at which the curvature changes sign, so it is not necessary to be a stationary point.
What is the second derivative used for?
The second derivative f"(x) can be used to determine the concavity and the points of inflection of f(x). If f"(x) is positive, then the graph of f(x) is concave up. If f"(x) is negative, then f(x) is concave down. If f"(x) is equal to zero, then f(x) has a point of inflection at that point.
What does the derivative at a point mean?
The derivative at any point in a curve is equal to the slope of the line tangent to the curve at that point. Doing it in terms of the actual expression of the curve, find the derivative of the curve, then plug the x-value of the point into the derivative to find the derivative at that point.
What Is Point Of Inflection?
An inflection point is a point on a curve at which the sign of the curvature (i.e., the concavity) changes.
How do you find the inflection point on an equation that has a variable to the third power?
Take the second derivative, set that equal to zero, and solve for x. All the possible x's you get from that are the x coordinates for the inflection points. To get the y coordinates, plug the x's back into the original equation and solve for y. I say x's because sometimes there will be multiple outputs. But since you're dealing with an equation to the third power, the second derivative's power will only be to the first, giving you only one x (one inflection point).
How do you trace a curve in differential calculus?
To trace a curve using differential calculus, you use the fact that the first derivative of the function is the slope of the curve, and the second derivative is the slope of the first derivative. What this means is that the zeros (roots) of the first derivative give the extrema (max or min) or an inflection point of the function. Evaluating the first derivative function at either side of the zero will tell you whether it is a min/max or inflection point (i.e. if the first derivative is negative on the left of the zero and positive on the right, then the curve has a negative slope, then a min, then a positive slope). The second derivative will tell you if the curve is concave up or concave down by evaluating if the second derivative function is positive or negative before and after extrema.
What is the critical point called?
The critical point is called the point at which a function's derivative is zero or undefined. At this point, the function may have a local maximum, minimum, or an inflection point.
Is critical point also an inflection point?
no, a critial point is where the slope (or the derivitive) is 0. the inflection point is when the graph switches from concave up to concave down or vice versa
Point of inflection in continuous beams?
point of zero moment
