Dog to the Dog

In exercises 1-6, find dydx :

1) 3y3+x2=5 2) x2y+2xy2=x+y

3) x4+y4=1 4) y-32+x32=1

5) y+xy=1 6) sinx+y=x+cos⁡(y)

Find the equations of the tangent and normal lines at the given point:

7) x23+y23=5, (1, 8)

Find all x-values where the tangent line to the graph is a) horizontal and b) vertical.

8) 3x2+4y2+3xy=24

Find d2ydx2 :

9) Problem 1. 10) y3-32x2=1

Solve the Related Rates Problem:

11) A road perpendicular to a highway leads to a farmhouse located 1 mile away. A car travels past the farmhouse at a speed of 60 mph. How fast is the distance between the automobile and the farmhouse increasing when the car is 3 miles past the intersection of the highway and road?

12) A plane passes directly above a radar station at an altitude of 6 miles. If the plane's speed is 500 mph, how fast is the distance between the plane and the radar station changing a half hour later?

13) A hot air balloon rising vertically is tracked by and observer located 2 miles from the lift-off point. At a certain moment, the angle between the observer's line of sight and the horizontal is π/5, and it is changing at a rate of 0.2 rad/min. How fast is the balloon rising at that moment?

14) A water tank in the shape of an inverted right circular cone of radius 300 cm and height 500 cm leaks water from the vertex at a rate of 10 cm3/min. Find the rate at which the water level is decreasing when the height is 200cm. Use the following formula for volume of a right circular cone: V=13πr2h

15) A balloon shaped like a sphere is leaking air at a rate of 5 cm3/sec when the volume is 400 cm3. Find the rate of change in the surface area of the balloon at that moment. Use the following formulas for Volume and Surface area of a sphere: V=43πr3 ; SA=4πr2