Exploring Behaviors of Implicit Relations Quiz

Let C be the curve defined by x^2⁢y=4. Which of the following statements is true of curve C at the point (2,1)?

It is decreasing and concave up because y′<0 and y″>0.

Let C be the curve defined by xy=−4. Which of the following statements is true of curve C at the point (−2,2) ?

It is increasing and concave up because y′>0 and y″>0.

Consider the curve in the x⁢y-plane defined by x^2−y^2/5=1. It is known that dy/dx=5⁢x/y and d2⁢y/d⁢x2=−25/y^3. Which of the following statements is true about the curve in Quadrant IV ?

The curve is concave up because d2y/dx2>0.

Consider the curve defined by x^2/16−y^2.9=1. It is known that dy/dx=9⁢x/16⁢y and d2⁢y/dx2=−81/16⁢y^3. Which of the following statements is true about the curve in Quadrant IV?

The curve is concave up because d2⁢y/d⁢x2>0.

Which of the following describes the x-coordinates of the points on the curve cos⁡x=e^y in the xy-plane where the curve has a horizontal tangent line?

The values 2npi for all integers n

Which of the following describes the y-coordinates of the points on the curve e^x=sin⁡y in the x⁢y-plane where the curve has a vertical tangent line?

The values pi/2+2npi for all integers n

At how many points on the curve x^2/3+y^2/3=9 in the xy-plane does the curve have a tangent line that is horizontal?

Two

At how many points on the curve x^2/5+y^2/5=1 in the xy-plane does the curve have a tangent line that is horizontal?

Two

If x^2y-x/y=-2, then dy/dx=

y-2xy^3/x+x^2y^2

Consider the curve in the xy-plane defined by y√y^3+1=x for y>-1. For what value of y, y>-1, does the derivative of y with respect to x not exist?

y=(-2/5)^1/3

Given the curve y2^y=x in the xy-plane, for what value of y, if any, does the derivative of y with respect to x not exist?

y=-1/ln2

If xy^2+x^2/y=5, then dy/dx=

-y(y^3+2x)/x(2y^3-x)