AP Calculus BC
2.3 – 2.5 Practice
1. Find
dy
4x  3
if y 
dx
2x  1
a)
10
b)
4 x  3
2.) Let f ( x) 
2
 10
c)
4 x  3
2
10
2 x  1
2
1 4
x  4x2
2
I. Find all the points where f has horizontal tangents.
II. Find an equation of the tangent line at x =1
III. Find an equation of the normal line at x = 1
3.) Which of the following gives
a) 4sin 3  3x  cos  3x 
c) 12sin  3x  cos  3 x 
dy
for y  sin 4 3 x 
dx
b) 12sin 3  3 x  cos  3x 
d) 12sin 3  3 x 
e) -12sin 3  3 x  cos  3 x 
d)
 10
2 x  12
e) 2
4.) Which of the following give y  for y  cos( x)  tan( x)
a ) -cosx  2sec2 x tan x
b) cosx  2sec 2 x tan x
c) -sinx  sec 2 x
d) -cosx  sec 2 x tan x
e) cosx  sec2 x tan x
5.) Which of the following gives
a) 11/2
b) 7/2
6.) Which of the following gives
dy
at the point (1,4) if x 3  2 xy  9
dx
c) 3/2
d) -7/2
e) -11/2
dy
if y  cos 3 3x  2
dx
a)  9cos 2  3x  2  sin  3 x  2 
b) -3cos 2 3 x  2  sin  3x  2 
c) 9cos 2  3x  2  sin  3 x  2 
d) -cos 2 3 x  2 
e) -3cos 2  3x  2 
7.) A curve in the xy-plane is defined by xy 2  x 3 y  6
a) Find
dy
dx
b) Find an equation for the tangent line on the curve at the point (1, -6).
c) Find an equation of the normal line at the point (1, -6).
8.) Find 𝑓 (6) (𝑥) of 𝑓(𝑥) = cos 𝑥
𝑑𝑦
9.) Find 𝑑𝑥 of cos2 𝑥 + cos2 𝑦 = cos(2𝑥 + 2𝑦)
𝑑𝑦
10.) Find 𝑑𝑥 of x2 – 5y3 = 0
11.) Find y  of xy3 + 3x – 4y2 = 12
𝑑𝑦
12.) Find 𝑑𝑥 of 2 sin(x) + cos (y) = 4
𝑑𝑦
13.) Find 𝑑𝑥 of tan (y) = sec (x) + x4y
𝑑𝑦
14.) Find 𝑑𝑥 of 3xcos (3y4) = sin (x)
Related Rates:
1. Given 𝑥 2 + 𝑦 2 = 25. Find
𝑑𝑦
𝑑𝑡
when 𝑥 = 3, 𝑦 = 4 and
𝑑𝑥
𝑑𝑡
=8
2. A hot-air balloon rising straight up from a level field is tracked by a range finder 500 ft from the lift-off
𝜋
point. At the moment the range finder’s elevation angle is 4 , the angle is increasing at the rate of 0.14 radians
per minute. How fast is the balloon rising at that moment?
𝑐𝑚3
3. Air is being pumped into a spherical balloon at a rate of 5 𝑚 . Determine the rate at which the radius of the
balloon is increasing when the diameter of the balloon is 20 𝑐𝑚.
𝑓𝑡 3
4. Sand is being dropped at the rate of 10 𝑚𝑖𝑛 onto a conical pile. If the height of the pile is always twice the
base radius, at what rate is the height increasing when the pile is 8 ft high?
𝑓𝑡 3
5. A water tank in the form of an inverted cone is being emptied at the rate of 6 𝑚𝑖𝑛. The altitude of the cone is
24 t and the base radius is 12 ft. Find how fast the water level is lowering when the water is 10 ft deep.
6. A man 6 feet tall walks at a rate of 5 feet per second away from a light that is 15 feet above the ground. When
he is 10 feet from the base of the light, (a) at what rate is the tip of his shadow moving? (b) at what rate is the
length of his shadow changing?
𝑓𝑡 3
7. A tank of water in the shape of a cone is leaking water at a constant rate of 2 ℎ𝑜𝑢𝑟. The base radius of the tank
is 5 ft and the height of the tank is 14 ft. At what rate is the radius of the water in the tank changing when the
depth of the water is 6 ft?
8. David and Angela start at the same point. At time t = 0, Angela starts running 30ft/sec north, while David
starts running 40ft/sec east. At what rate is the distance between them increasing when they are 100 feet apart?
9. Water is being poured into an inverted cone (has the point at the bottom) at the rate of 4 cubic centimeters per
second. The cone has a maximum radius of 6cm and a height of 30 cm. At what rate is the height increasing
when the height is 3cm?