Consider the implicit relation x y+y^ 2 =

MATHEMATICS · Multi-part · CBSE · Mathematics · Continuity and Differentiability · Continuity and Differentiability

Problem

Consider the implicit relation xtan⁡y+y2=π.x\tan y+y^{2}=\pi. All ordered pairs (x,y)(x,y) mentioned below satisfy this equation. (a) Derive an explicit expression for dydx\dfrac{dy}{dx} in terms of xx and yy. Simplify as much as algebraically convenient. (b) Obtain a formula for the second derivative d2ydx2\dfrac{d^{2}y}{dx^{2}} expressed entirely in terms of xx and yy. Show all steps in your differentiation. (c) The point Q ⁣(π4,π4)Q\!\left(\dfrac{\pi}{4},\dfrac{\pi}{4}\right) is on the curve. Evaluate dydx\dfrac{dy}{dx} and d2ydx2\dfrac{d^{2}y}{dx^{2}} at QQ. Write each value in simplest exact form. (d) Use your result from part (c) to determine whether the graph of the relation is concave up or concave down at QQ. Justify your conclusion clearly.

Related theory

Browse the problem library