Advanced Maths:Geometric Applications of Differentiation

From Biki

Increasing ,decreasing and stationary at a point If f’(x)is +ve then f(x) is increasing If f’(x)is –ve then f(x) is decreasing If f(x) = 0 then f(x) is stationary Stationary points and turning points

Critical values E.g. Find the critical value of y= 1/(x(x-4)) CV x≠0,4 Higher derivatives and concavity and points of inflection E.g. differentiate y=x^10 y^'=〖10x〗^9 y^=90x^8 If f’’(x)>0,f(x) is concaving upwards and it’s a min If f’’(x)<0,f(x) is concaving downwards and it’s a max If f’’(x)=0 and changes concavity then there is a point of inflection

Curve sketching using calculus 1.domain 2.range 3.intercepts and signs 4.asymtotes 5.the first derivative 6.the second derivative 7.odd/even function Global maximum and global Minimum E.g. state the global max and min of y=x^2,-2≤x≤2 y≤4 y^'= 2x Stat pts(0,0) Global min and max at 0,4

Applications of maximums and minimums You MUST always test you stationary points in any question that requires you to find the max and min E.g. a)if x + y=8,express p=x^2+y^2 In terms of x only y=8-x p=x^2+(8-x)^2 =x^2+64+x^2-16x =2x^2-16x+64 b)find p’ and hence find the value of x for which p obtains it minimum value p^'=4x-16 0=4x-16 〖 min〗⁡〖when p^'=0〗 4x=16 x=4 p^=0 Therefore x=4 is a min Maximisation and minimisation in geometry