Practise implicit-differentiation questions by finding derivatives from equations and reasoning about the behavior of implicitly defined functions.
Given the following differential equation
tart fraction numerator d y over denominator d x end fraction equals x minus
2 y plus 1
(b) Find the second derivative,
et fraction numerator d squared y over denominator d x squared end
fraction
, in terms of x and y. The region in the x y-plane where all the solution
curves to the differential equation are concave down can be expressed
as a linear inequality. Find this region.
(b)
art fraction numerator d squared end exponent y over
denominator d x squared end exponent end fraction equals 1
minus 2 start fraction numerator d y over denominator d x end
fraction equals 1 minus 2 left parenthesis x minus 2 y plus 1
right parenthesis equals 4 y minus 2 x minus 1
The solution curves will be concave down when
- start fraction numerator d squared y over denominator d x
squared end fraction less than 0
4 y minus 2 x minus 1 less than 0 rightwards double arrow y less than 1 over 2
x plus one-fourth