Flashcards
Nature of the Derivative
Cards in this deck (13)
- 1This graph is for next flashcards so look at it.....
- 2What is represented by the notation d^2y/dx^2 ?The 2nd derivative (gradient of the gradient)
- 3What is the gradient at A and B and what are they called?dy/dx = 0
Stationary points - 4What is a local minimum?dy/dx = 0
d^2y/dx^2 > 0 - 5What is a local maximum?dy/dx = 0
d^2y/dx^2 < 0 - 6What is a point of inflection?dy/dx = 0
d^2y/dx^2 = 0
BUT if d^2y/dx^2 = 0 then it may not be a point of inflection (could also be local min or max) - 7IF given a function, to find the coordinates of any stationary point what must we do?Solve dy/dx = 0 and state (x,y)
- 8If we are asked to determine the nature of stationary points, what must we find?The local maximum, the local minimum and the points of inflection
- 9What is point A?A point of inflection (looks like its going to turn but doesnt and keeps going down)
- 10What is point B?A local minimum
- 11If asked to find coordinates of a local min/max when given a function such as y=15x^(3/2)-x^(5/2) , what should we do?find dy/dx, set it to 0, then factorize/solve to find x coordinate, then sub the x coordinate into the equation to find the y coordinate. (Anser to this example is (9,162) )
- 12How do we find the equation of the tangent line to y=x^3+x at x=1 ?Plug in x=1: 1^3+1 = 2 (this is y1)
Find gradient of tangent at x=1: dy/dx(x=1):
3 x 1^2 +1 = 4 (this is m)
As we are finding x=1, x1=1
Put into y-y1=m(x-x1) ;
y-2=4(x-1) , change format if asked - 13How to we find the equation of the normal line to y=5x+x^4 at x=1Plug in x=1: 5x1 +1^4 = 6
Find gradient of tangent at x=1: dy/dx(x=1):
5+4x1^3 = 9
Gradient of normal = -1/9, as -1/9 x 9 = -1
Put into y-y1=m(x-x1) ;
y-6=-1/9(x-1) , change format if asked