Homework SourseConsider the graph defined by xtt24 ytt2t3 i Compute dydx ii
Consider the graph defined by: x(t)=t^2+4, y(t)=t^2+t^3.
i. Compute dy/dx.
ii. Compute d^2y/dx^2.
iii. For what t is the curve concave up (on the x-y plane)?
i. Compute dy/dx.
ii. Compute d^2y/dx^2.
iii. For what t is the curve concave up (on the x-y plane)?
Solution
x(t)=t^2+4, y(t)=t^2+t^3 dx/dt = 2t, dy/dt = 2t+3t^2 dy/dx = dy/dt / dx/dt dy/dx = (2t+3t^2)/2t dy/dx = 3t/2 + 1......(1) d^2y.dx^2 = 3/2 dt/dx d^2y/dx^2 = 3/2 * 1/2t d^2/dy^2 = 3/4t.......(2) for concave up. d^2y/dx^2 > 0 3/4t > 0 hence,t > 0.....(3)
