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How do you check if a function is convex or concave?

A convex function has an increasing first derivative, making it appear to bend upwards. Contrarily, a concave function has a decreasing first derivative making it bend downwards.

What is the volume of solid of revolution when rotated along the x-axis?

When revolved along x-axis, the volume is given as \inta^b y^2 dx . Explanation: Volume is generated when a 2d surface is revolved along its axis.

How can we say a function is convex?

A function f : Rn R is convex if and only if the function g : R R given by g(t) = f(x + ty) is convex (as a univariate function) for all x in domain of f and all y Rn. (The domain of g here is all t for which x + ty is in the domain of f.)

How do you determine if a function is convex or concave Hessian?

A function is convex if the Hessian matrix of that function is positive semi-definite. When the Hessian matrix is positive definite, the function is strictly convex. To test the concavity of a function, compute the Hessian matrix and prove it is negative semi-definite.

How do you find the volume of a solid rotated about a line?

3:20 8:59 Volume of a Solid Rotated About Any Line - YouTube YouTube Start of suggested clip End of suggested clip My inner curve. Or do my inner curve minus the axis of rotation subtract the axis of rotation fromMoreMy inner curve. Or do my inner curve minus the axis of rotation subtract the axis of rotation from the inner curve. And square it then we can get the circle of our hole.

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How do you find the volume of a solid rotated about the X axis?

3:25 19:59 Disk Washer Method - Calculus - YouTube YouTube Start of suggested clip End of suggested clip So the volume is going to be from. You have to integrate the function from a to b. And its going toMoreSo the volume is going to be from. You have to integrate the function from a to b. And its going to be r squared of x dx thats how you find the volume of the solid rotated about the x-axis.

How do you define Y as a function of X?

0:14 1:59 Determine if an equation defines y as a function of x - YouTube YouTube Start of suggested clip End of suggested clip Theres. Only one value for y. So this one is a yes. Now what about over here well if I want to getMoreTheres. Only one value for y. So this one is a yes. Now what about over here well if I want to get Y by itself. Im going to have to subtract x squared.

Which relation is a function of X?

A function is a relation in which each input has only one output. In the relation , y is a function of x, because for each input x (1, 2, 3, or 0), there is only one output y.

Which functions are concave?

A differentiable function f is (strictly) concave on an interval if and only if its derivative function f is (strictly) monotonically decreasing on that interval, that is, a concave function has a non-increasing (decreasing) slope.

How to find the volume of a solid rotated about the x-axis washer method?

3:25 19:59 Disk Washer Method - Calculus - YouTube YouTube Start of suggested clip End of suggested clip So the volume is going to be from. You have to integrate the function from a to b. And its going toMoreSo the volume is going to be from. You have to integrate the function from a to b. And its going to be r squared of x dx thats how you find the volume of the solid rotated about the x-axis.

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How do you check if a function is convex or concave?

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