1. [latex]x^2-y^2=4[/latex]

2. [latex]6x^2+3y^2=12[/latex]

3. [latex]x^2 y=y-7[/latex]

4. [latex]3x^3+9xy^2=5x^3[/latex]

5. [latex]xy- \cos (xy)=1[/latex]

6. [latex]y\sqrt{x+4}=xy+8[/latex]

7. [latex]−xy-2=\dfrac{x}{7}[/latex]

8. [latex]y \sin(xy)=y^2+2[/latex]

9. [latex](xy)^2+3x=y^2[/latex]

10. [latex]x^3 y+xy^3=-8[/latex]

11. [T]  [latex]x^4 y-xy^3=-2, \,\,\, (-1,-1)[/latex]

12. [T]  [latex]x^2 y^2+5xy=14, \,\,\, (2,1)[/latex]

Show Solution

[latex]y=-\frac{1}{2}x+2[/latex]

13. [T]  [latex]\tan (xy)=y, \,\,\, (\frac{\pi}{4},1)[/latex]

14. [T]  [latex]xy^2 + \sin(\pi y)-2x^2=10, \,\,\, (2,-3)[/latex]

Show Solution

[latex]y=\large \frac{1}{\pi +12}x-\frac{3\pi +38}{\pi +12}[/latex]

15. [T]  [latex]\frac{x}{y}+5x-7=-\frac{3}{4}y, \,\,\, (1,2)[/latex]

16. [T]  [latex]xy+ \sin (x)=1, \,\,\, (\frac{\pi}{2},0)[/latex]

Show Solution

[latex]y=0[/latex]

17. [T] The graph of a folium of Descartes with equation [latex]2x^3+2y^3-9xy=0[/latex] is given in the following graph.

1. Find the equation of the tangent line at the point [latex](2,1)[/latex]. Graph the tangent line along with the folium.
2. Find the equation of the normal line to the tangent line in a. at the point [latex](2,1)[/latex].

18. For the equation [latex]x^2+2xy-3y^2=0[/latex],

1. Find the equation of the normal to the tangent line at the point [latex](1,1)[/latex].
2. At what other point does the normal line in a. intersect the graph of the equation?

19. Find all points on the graph of [latex]y^3-27y=x^2-90[/latex] at which the tangent line is vertical.

20. For the equation [latex]x^2+xy+y^2=7[/latex],

1. Find the [latex]x[/latex]-intercept(s).
2. Find the slope of the tangent line(s) at the [latex]x[/latex]-intercept(s).
3. What does the value(s) in b. indicate about the tangent line(s)?

21. Find the equation of the tangent line to the graph of the equation [latex]\sin^{-1} x+\sin^{-1} y=\frac{\pi}{6}[/latex] at the point [latex](0,\frac{1}{2})[/latex].

22. Find the equation of the tangent line to the graph of the equation [latex]\tan^{-1}(x+y)=x^2+\frac{\pi}{4}[/latex] at the point [latex](0,1)[/latex].

23. Find [latex]y^{\prime}[/latex] and [latex]y''[/latex] for [latex]x^2+6xy-2y^2=3[/latex].

24. [T] The number of cell phones produced when [latex]x[/latex] dollars is spent on labor and [latex]y[/latex] dollars is spent on capital invested by a manufacturer can be modeled by the equation [latex]60x^{\frac{3}{4}}y^{\frac{1}{4}}=3240[/latex].

1. Find [latex]\frac{dy}{dx}[/latex] and evaluate at the point [latex](81,16)[/latex].
2. Interpret the result of a.

25. [T] The number of cars produced when [latex]x[/latex] dollars is spent on labor and [latex]y[/latex] dollars is spent on capital invested by a manufacturer can be modeled by the equation [latex]30x^{\frac{1}{3}}y^{\frac{2}{3}}=360[/latex].

(Both [latex]x[/latex] and [latex]y[/latex] are measured in thousands of dollars.)

1. Find [latex]\frac{dy}{dx}[/latex] and evaluate at the point [latex](27,8)[/latex].
2. Interpret the result of a.

26. The volume of a right circular cone of radius [latex]x[/latex] and height [latex]y[/latex] is given by [latex]V=\frac{1}{3}\pi x^2 y[/latex]. Suppose that the volume of the cone is [latex]85\pi \, \text{cm}^3[/latex]. Find [latex]\frac{dy}{dx}[/latex] when [latex]x=4[/latex] and [latex]y=16[/latex].

27. Find an equation for the surface area of the rectangular box, [latex]S(x,y)[/latex].

28. If the surface area of the rectangular box is 78 square feet, find [latex]\frac{dy}{dx}[/latex] when [latex]x=3[/latex] feet and [latex]y=5[/latex] feet.

29. [latex]x= \sin y[/latex]

30. [latex]x= \cos y[/latex]

31. [latex]x= \tan y[/latex]