Implicit Differentiation Review
1. Given x2 - xy + y2 = 21, find:
a) dy/dx
b) Find the y-coordinate(s) when x = 1.
c) Find the slope of the tangent line at the points in (b).
d) Write the equation(s) of the tangent line(s) to the curve at x = 1.
e) The coordinates of the points where the tangent line to the curve is horizontal.
2. The function f is differentiatiable for all real numbers. The point (3, ¼) is on the graph of
y = f(x), and the slope at each point (x, y) on the graph is given by
a) Find
𝑑2 𝑦
𝑑𝑥 2
𝑑𝑦
𝑑𝑥
= 𝑦 2 (6 − 2𝑥).
and evaluate it at the point (3, ¼).
b) Is the graph concave up, concave down, or neither at (3, ¼)? Explain.
3. If tany + x3 = y2 + 1 and
A. -6
B. -2.5
𝑑𝑥
𝑑𝑡
= −2, what is the value of dy/dt at the point (1, 0)?
C. 0
D.
1
2(𝑐𝑜𝑠1)2
E. 6
Related Rates Review
4. All edges of a cube are expanding at a rate of 3 centimeters per second. How fast is the
volume changing when each edge is 10 cm?
5. A conical tank (with vertex down) is 12 feet across and 10 feet deep. If water is flowing
into the tank at a rate of 5 cubic feet per minute, find the rate of change of the depth of the
water when the water is 6 feet deep. (V = 1/3πr2h)
6. Air is blown into a spherical balloon. How fast is the radius changing when the air is
blown into the balloon at the rate of 10 cm3/s at the instant the radius is 3 cm? (V= 4/3πr3)
7. A rocket, rising vertically, is tracked by a radar station that is on the ground 3000 feet
from the launching pad. At what rate is the angle of elevation changing when the rocket is
4000 feet up and rising vertically at 5000 feet per second?
8. A pebble is dropped into a calm pond, causing ripples in the form of concentric circles.
The radius “r” of the outer ripples is increasing at a constant rate of 1 foot per second.
When the radius is 4 feet, at what rate is the total area “A” of the disturbed water
changing?
9. The radius of a sphere is increasing at a constant rate of 0.04 cm/s.
a) At the time when the radius of the sphere is 10 cm, what is the rate of increase of its
volume? (V = 4/3πr3)
b) At the time when the volume of the sphere is 36π cm3, what is the rate of increase of the
area of a cross section through the center of the sphere?
10. The height of a conical sand pile is always twice the raidus. If sand is being added to
the pile at a rate of 30π cm3/min, how fast is the height of the pile increasing when the
circumference of the base of the sand pile is 120π cm? (V = 1/3πr2h)
A.
1
𝑐𝑚/𝑚𝑖𝑛
120𝜋
B.
1
𝑐𝑚/𝑚𝑖𝑛
120
C.
2
𝑐𝑚/𝑚𝑖𝑛
15
1
D. 4 𝑐𝑚/𝑚𝑖𝑛
E. None of these
11. Two cars are converging on a point P as they drive at right angles to each other. Car A
is traveling at 60mph and car B is traveling at 50mph. At the instant when car A is 12
miles from the point P and car B is 10 miles from the point P, at what rate is the distance
between the cars decreasing?
A. 55mph
B. 55.455mph
C. 76.882mph
D.78.102mph
E. none of these
12. A regular tetrahedron is a solid figure with four faces which are congruent equilateral
triangles. If the lengths of the edges of the tetrahedron are increasing at the constant rate
of 2 cm/min, then how fast is the surface area of the tetrahedron increasing when the
edges are each 12 cm in length?
A. 8 cm2/min
B. 20.785 cm2/min
D. 48 cm2/min
E. 83.138 cm2/min
C. 41.569 cm2/min