I'm really not confident with differential calculus, and need to get a grip on it before my assessment?

Question

Consider the function f(x) = |x^2 - 4|

The function has three points which are A = (0,4), B = (-2,0) & C = (2,0).

a) What would be an appropriate scale on the axes.?

b) I'm assuming the coordinates of the points A,B,& C are as I've written them above ?

c) Find the derivatives at these points and explain your answers

Consider the function f(x) = |x^2 - 4|

The function has three points which are A = (0,4), B = (-2,0) & C = (2,0).

a) Write in the scale on the axes.?

b) I'm assuming the coordinates of the points A,B,& C are as I've written them above ?

c) Find the derivatives at these points and explain your answers

Answer 1

you can define a function called signf(x) whose value is +1 if f(x) is positive .Is -1 if f(x) is negative and0 if f(x) =0
Its derivative is always 0 except at 0 where its doesn´t exist.
So if you have a function If(x)IYou can write it as f(x) *signf(x)
and its derivative would be
f´(x)sign(f(x) + f(x) [signf(x)]´ The second term is always zero except if f(x)=0

In your case I f(x)I =(x^2-4)*sign(x^2-4) and its derivative

If(x)I´= 2x*sign(x^2-4) +(x^2-4)*[sign(x^2-4)]´
At x=0 x^2-4 = -4 so sign is - 1 and the derivatve=2*0*(-1)+0=0
A tx=-2 f(x)=0 and the derivative doesn´t exist.The same at x=2

Both are called angular points of the graph

Answer 2

a) scale? Not sure of your exact meaning, but look at your points A, B,C - the left most is -2, rightmost is 2, largest is 4, smallest is 0. Try plotting the graph a bit to the left of -2 and a bit to the right of 2. So, put -3 and 3 in for x and see what happens.

b) Yep, those are correct.

c) You can do this without really taking the derivative. Look at the graph. At point A you have a local maximum, so the derivative is 0 there. At points B and C you have cusps, sharp turning points due to the absolute value, so the derivative doesn't exist there.

Answer 3

I got a 40% on my calc test on friday........i'm going to kill myself