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How do you know if the second derivative of an equation is a maximum or a minimum?

If the second derivative of an equation is positive (d 2 y /dx 2 > 0), we can see that this point on a curve is a minimum. This is because, where the first derivative finds the gradient of a curve (how th...
JG
Answered by Jessica G. • Maths tutor
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Use integration by parts to find the integral of ln x by taking ln x as the multiple of 1 and ln x

For integration by parts, the integral is uv - ∫ u' v dx. First we take u = ln x and v' = 1. While we could have u and v' be the opposite at this stage, it becomes apparent later on that we can't do this bec...
JC
Answered by Jack C. • Maths tutor
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(The question is too long so it's marked at the top of the answer space, sorry for any inconveniences)

Question: Particle P of mass m and particle Q of mass km are moving in opposite directions on a smooth horizontal plane when they collide directly. Immediately before the collision the speed of P is 5u and t...
XW
Answered by Xudong W. • Maths tutor
5162 Views

Integrate (3x^2 - (1/4)x^-2 + 3) dx

3x 2 -(1/4)x -2 +3 Split the problem into three separate terms and integrate independently.With any term where x is to the power of something, divide the coefficient by the value of the power plus 1 .e.g. 3x...
LL
Answered by Lucy L. • Maths tutor
5095 Views

Find the integral of 4/(1-x^2) dx:

The first thing to notice here is that the denominator of the integrand is a case of 'difference of two squares'. The integral, which I will call I, can be rewritten as the integral of 4/((1+x)(1-x)) dx. If ...
JP
Answered by Jemima P. • Maths tutor
6225 Views