AP Calculus AB Fun 4 E Justify Conclusions About the Behavior of an Implicitly Defined Function Based on Evidence From Its Derivatives Questions

Practise implicit-differentiation questions by finding derivatives from equations and reasoning about the behavior of implicitly defined functions.

Exam points

differentiate an implicitly defined equation with respect to x
use derivative evidence to justify behavior of an implicitly defined function

Given the following differential equation $\frac{dy}{dx}=x-2y+1$ .

Find the second derivative, $\frac{d^2y}{dx^2}$ , 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.

$\frac{d^2y}{dx^2}=1-2\frac{dy}{dx}=1-2(x-2y+1)=4y-2x-1$ . The solution curves will be concave down when $\frac{d^2y}{dx^2}<0$ . $4y-2x-1<0\Rightarrow y<\frac12x+\frac14$ .

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AP Calculus AB Fun 4 E Justify Conclusions About the Behavior of an Implicitly Defined Function Based on Evidence From Its Derivatives Questions question 1