Math 180 #003
UNM
Summer 2012
Elements of Calculus I: Worksheet 5
1. Consider the equation x 2  5 y 2  36 . Suppose that y  y ( x ) .
dy
dy
(a) Find
.
(b) Find
.
dx
dx ( x , y )(4,2)
2. Consider the equation x 2  5 y 2  36 . Suppose that that y  y (t ) and x  x(t ) .
dy
dy
(a) Find
.
(b) If x(3)  4 , y (3)  2 , and x '(3)  5 , find
.
dt
dt t 3
3. Consider the elliptical equation: x 2  xy  y 2  4 .
(a) Find the coordinates of the two points on the ellipse that intersect the line y  x .
(b) Show that the tangent lines to the ellipse at the two points in part (a) are parallel.
4. Consider the graph of the equation: x 4  2 x 2 y 2  y 4  4 x 2  4 y 2 .
(a) Show that the point

1
2

6, 12 2 is on the graph.
(b) Find the slope of the tangent line to the graph at the point

1
2

6, 12 2 .
5. Consider the curve x 2 y  ay 2  b , where a and b are constants. Suppose the curve
satisfies the following two conditions:
(i) The point (1,1) is on its graph.
(ii) The equation of the tangent line to the curve at (1,1) is 4 x  3 y  7 .
Find the values of a and b.
6. If y  y ( x ) and f ( x)  g ( y )  0 , then find dy / dx .
7.
(a) Write the formula for the volume of a cylinder.
(b) Find the rate of change of volume with respect to height if the radius is constant.
(c) Find the rate of change of volume with respect to radius if the height is constant.
(d) Find the rate of change of volume with respect to time if the radius changes with
time and the height is constant.
(e) Find the rate of change of volume with respect to time if the height changes with
respect to time and the radius is constant.
(f) Find the rate of change of volume with respect to time if both the radius and the
height change with respect to time.
(g) Find the rate of change of volume in part (d) when the height is 3 m, the volume
is 27 m3, and the radius is decreasing at a rate of 2 m/s.
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8. A coffee pot in the form of a circular cylinder of radius 4 in. is being filled with water
flowing at a constant rate. If the water level is rising at the rate of 0.4 in./sec, what is the
rate at which water is flowing into the coffee pot?
9. Two ships leave a port at the same time in the morning, one headed north and the other
headed east. At noon, the northbound ship was 40 miles from port and sailing at 30 mph,
while the eastbound ship was 30 miles from port and sailing at 20 mph. How was the
distance between the two ships changing at that time?
10. The width of a rectangle is increasing at a rate of 3 in/sec and its length is increasing at a
rate of 4 in/sec. At what rate is the area of the rectangle increasing when its width is 5 in
and its length is 6 in?
11. The body mass index, or BMI, is a ratio of a person's weight divided by the square of his
or her height. If b(t ) represents the BMI of a person at age t, then:
w(t )
b(t ) 
,
2
 h(t )
where w(t ) is the weight in kilograms at age t and h(t ) is the height in meters at age t.
(a) At age 12, a boy's weight was 50 kg and his height was 1.5 m. Find his BMI at
age 12.
(b) In addition to the information in part (a), suppose that the boy's weight was
increasing at a rate of 7 kg/year and his height was increasing at a rate of 5
cm/year at age 12. Find the rate of change of his BMI at age 12.
12. The demand equation to sell x units of product when the price is p dollars is
p  2 x  xp  38 , where both x and p depend on time. Find the rate at which the sales are
changing at the time 4 units are sold at a price of $6 when the price is decreasing at a rate
of $0.40/week.
13. The volume of a spherical cancerous tumor is given by V   x 3 / 6 , where x is the
diameter in millimeters. If the tumor is growing at a rate of 0.4 millimeters per day, then
find the rate at which the volume of the tumor is changing at the time the radius of the
tumor is 5 millimeters.
14. An airplane flying 390 ft/sec at an altitude of 5000 feet flew directly over a person in a
straight line. How fast is the distance from the person to the airplane changing at the
moment the airplane is 13,000 feet away from the person.
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