Implicit Differentiation and Related Rates
Practice
4.1 The graph of the 
is a rotated ellipse.
a) Using implicit differentiate for 
b) Show that the tangent lines to this curve at the two points where it intersects the x-axis are parallel.
At any time that y = 0 the slope of this graph will be 2, thus the tangent lines are parallel.
c) 
4.2 A crane with a 30ft. arm is rotating downward at the rate of .1 radians/sec. How fast is the vertical distance changing when 
Negative because the arm is rotating downwards so the height is decreasing
4.3 A cone of height 15 inches and a radius of 5 inches is being filled at the rate of 9in³/sec. How fast is the height changing at the instant height = 6 inches?
4.4 x and y vary inversely: xy = 240. x is changing at the rate of 3 units/sec. How fast is y changing when 
units?
4.5 A right circular cone (the cone’s height is the same as the radius of the base. In this problem the cone fits perfectly inside the hemisphere) and a hemisphere have the same base. The combined surface area of the hemisphere and its base is increasing at the rate of 18 in²/sec. At what rate is the volume of the cone changing when the radius of the common base is 4 inches.
Because it is a right cone h = r
