Courtesy Dr. Rosanna Pearlstein
Math 151 WIR 6: Sections 3.7, 3.8, 3.9, 3.10
1. A satellite completes one orbit of Earth along the equator at an altitude of 2000 Km every 2 h 8 min. Find
the velocity, speed and acceleration of the satellite at any given time. Note: Earth’s radius is
approximately 6600 Km.
2. A cannonball is fired from a cannon with a muzzle velocity of 1000 ft/s, inclined at an angle of 60° with
the horizontal. The position of the cannonball t seconds after the cannon is fired is given by
𝑥 = 500𝑡 and 𝑦 = 500√3𝑡 − 16𝑡 2 .
a. Compute the velocity, speed and acceleration of the cannonball at the time the ball hits the ground.
b. Compute the velocity, speed and acceleration of the cannonball when the ball reaches its maximum
height.
3. Find the point(s) on the curve 𝑥 = 𝑠𝑖𝑛2𝑡, 𝑦 = 𝑠𝑖𝑛𝑡, 0 ≤ 𝑡 ≤ 2π, where the tangent line is horizontal or
vertical.
4. Show that the curve 𝑥 = 𝑠𝑖𝑛𝑡, 𝑦 = 𝑠𝑖𝑛𝑡𝑐𝑜𝑠𝑡, 0 ≤ 𝑡 ≤ 2π, has two tangent lines at (0, 0) and find their
equations. Sketch the curve. Find all points where the tangent line is horizontal/vertical.
5. Solve parts (a) and (b) below.
a. Find equations of the tangent line(s) to the curve 𝑥(𝑡) = 3𝑡 2 + 1, 𝑦(𝑡) = 2𝑡 3 + 1 that pass through
the point (4, 3).
b. At which point(s) on the curve is the tangent line parallel to the line 2𝑥 + 5𝑦 + 4 = 0?
6. If a snowball melts so that its surface area decreases at a rate of 1 𝑐𝑚2 /𝑚𝑖𝑛, find the rate at which the
diameter decreases when the diameter is 10 cm.
7. A man starts walking north at 4 ft/s from a point P. Five minutes later a woman starts walking south at 5
ft/s from a point 500 ft due east of P. At what rate are the people moving apart 15 min after the woman
starts walking?
8. A kite 100 ft above the ground moves horizontally at a speed of 8 ft/s. At what rate is the angle between
the string and the horizontal decreasing when 200 feet of string have been let out?
9. Find the following derivatives:
a.
b.
𝑑50 [350 cos(−2𝑥)]
𝑑𝑥 50
𝑑50 √2𝑥
𝑑𝑥 50
Below is the solution to part b. Don’t look if you haven’t tried this on your own first!
Courtesy Dr. Rosanna Pearlstein
𝑑 50 √2𝑥
=?
𝑑𝑥 50
Let 𝒇(𝒙) = √2𝑥 = (𝟐𝒙)𝟏/𝟐 . Then
𝒇′(𝒙) = (𝟐𝒙)−𝟏/𝟐
𝟏
𝒇′′(𝒙) = −(𝟐𝒙)−𝟐−𝟏
𝟏
𝟓
𝒇′′′(𝒙) = +𝟑(𝟐𝒙)−𝟐−𝟏−𝟏 = +𝟑(𝟐𝒙)−𝟐
𝟏
𝟕
𝒇𝒊𝒗(𝒙) = −(𝟑)(𝟓)(𝟐𝒙)−𝟐−𝟏−𝟏−𝟏 = −(𝟑)(𝟓)(𝟐𝒙)−𝟐
𝟏
𝟗
𝒇𝒗(𝒙) = +(𝟑)(𝟓)(𝟕)(𝟐𝒙)−𝟐−𝟏−𝟏−𝟏−𝟏 = +(𝟑)(𝟓)(𝟕)(𝟐𝒙)−𝟐
𝟏
𝟏𝟏
𝒇𝒗𝒊(𝒙) = −(𝟑)(𝟓)(𝟕)(𝟗)(𝟐𝒙)−𝟐−𝟏−𝟏−𝟏−𝟏−𝟏 = −(𝟑)(𝟓)(𝟕)(𝟗)(𝟐𝒙)− 𝟐
… Do you see the pattern yet? What is the tenth derivative? Do it!
𝟏
𝒇(𝒕𝒆𝒏)(𝒙) = −(𝟑)(𝟓)(𝟕)(𝟗)(𝟏𝟏)(𝟏𝟑)(𝟏𝟓)(𝟏𝟕)(𝟐𝒙)−𝟐−𝟏−𝟏−𝟏−𝟏−𝟏−𝟏−𝟏−𝟏−𝟏
𝟏𝟗
= −(𝟑)(𝟓)(𝟕)(𝟗) … (𝟏𝟕)(𝟐𝒙)− 𝟐
So by the time we reach the fiftieth derivative…
𝒕𝒉𝒆𝒓𝒆 𝒂𝒓𝒆 𝟒𝟗 𝒐𝒇 𝒕𝒉𝒆 "−𝟏"
(𝒇𝒊𝒇𝒕𝒚)(𝒙)
𝒇
=
𝟏
−(𝟑)(𝟓)(𝟕)(𝟗)(𝟏𝟏)(𝟏𝟑)(𝟏𝟓)(𝟏𝟕) … (𝟗𝟕)(𝟐𝒙)−𝟐−𝟏−𝟏−𝟏−𝟏−𝟏−𝟏−𝟏−𝟏−⋯−𝟏
𝟗𝟗
= −(𝟑)(𝟓)(𝟕)(𝟗) … (𝟗𝟕)(𝟐𝒙)− 𝟐
Now, that was interesting was it not? Were you able to get the answer on your own?