Calculus Worksheet: Introduction to Related Rates
Recall Implicit Differentiation: Find
dy
if x 2  4 y 2  y  9
dx
Instead of finding dy/dx, we can sometimes be interested in finding dy/dt (the derivative of y with
respect to t)
1.
If
y
x , find dy / dt when x  4 given dx / dt  3.
2.
If
y  x , find dx / dt when x  25 given dy / dt  2.
3.
Given xy=4 and dx/dt=10, find dy/dt when x=8.
4.
Given xy=4 and dy/dt=-6, find dx/dt when x=1.
Why would we need to find dy/dt or dx/dt? The following problem was on the 2008 AP exam:
2008 AP® CALCULUS AB FREE-RESPONSE QUESTIONS
3. Oil is leaking from a pipeline on the surface of a lake and forms an oil slick whose volume
increases at a constant rate of 2000 cubic centimeters per minute. The oil slick takes the form of
a right circular cylinder
with both its radius and height changing with time.
(a) At the instant when the radius of the oil slick is 100 centimeters and the height is 0.5
centimeter, the radius is increasing at the rate of 2.5 centimeters per minute. At this instant,
what is the rate of change of the height of
the oil slick with respect to time, in centimeters per minute?
5.
The radius r of a circle is increasing at a rate of 2 inches per minutes. Find the rate of change of
the area when a) r=6 inches and b) r=24 inches.
6.
A spherical balloon is inflated so that its volume is increasing at a rate of 3 cubic feet per
minutes. How fast is the diameter of the balloon increasing when the radius is 1 ft?
7.
A spherical balloon is to be deflated so that its radius decreases at a constant rate of 15cm/min.
At what rate must air be removing when the radius is 9 cm?
8.
The radius r of a sphere is increasing at a rate of 2 inches per minutes. Find the rate of change
of the volume when a) r=6 inches and b) r=24 inches.
9.
The edge of a cube is growing at a rate of 4 inches per second. How fast is the volume of the
cube changing when the edge of the cube is 12 inches in length?
10. Air is being pumped into a spherical balloon so that its volume increases at a rate of 100 cubic
cm/second. How fast is the radius of the balloon increasing when the diameter is 50 cm?