#separator:tab
#html:false
#notetype:Basic
#deck:Math::Related rates
#tags:math calculus related-rates
#columns:Front	Back	Tags
#tags column:3
Two quantities are tied by an equation at every instant. What ties their rates?	The derivative of that equation with respect to time t.	idea
The one move in every related-rates problem?	Differentiate the equation that holds at every instant, then put in the numbers of the one instant asked about.	idea
Why do the numbers go in only after differentiating?	They hold at one instant only. Substituted first, the equation relates constants, whose derivative is 0, and the problem becomes nonsense like 2 = 0.	idea
Balloon: V = (4/3) pi r^3. Differentiate in t.	dV/dt = 4 pi r^2 dr/dt.	formulas
Why is the factor in dV/dt = 4 pi r^2 dr/dt the sphere's surface?	Growing r by a little adds a skin of volume about surface times thickness: the surface is what the volume gains per unit of radius.	idea
Circle: A = pi r^2. Differentiate in t.	dA/dt = 2 pi r dr/dt: the circumference times dr/dt.	formulas
Cube: V = a^3. Differentiate in t.	dV/dt = 3a^2 da/dt.	formulas
Sphere surface: S = 4 pi r^2. Differentiate in t.	dS/dt = 8 pi r dr/dt.	formulas
Ladder of length L: x^2 + y^2 = L^2. Differentiate in t.	2x dx/dt + 2y dy/dt = 0, so dy/dt = -x (dx/dt)/y.	formulas
Two travellers at right angles: x^2 + y^2 = z^2. Differentiate in t.	x dx/dt + y dy/dt = z dz/dt.	formulas
The seven questions for a related-rates problem?	What quantities appear? Which change? Which are fixed? What rate is given? What rate is requested? Should it be positive or negative? What equation relates the quantities?	questions
What is step one of every related-rates problem?	Draw the picture yourself and label every quantity.	questions
Why decide the sign of the answer before computing?	The picture tells you whether the quantity grows or shrinks. If the algebra disagrees, the algebra is wrong.	questions
A quantity that is decreasing has what kind of rate?	A negative one.	questions
Circle r = 5, dr/dt = 2: dA/dt?	20 pi, about 62.83 square meters per second.	problems
Balloon filled at 2 cm^3/s, r = 3: dr/dt?	1/(18 pi), about 0.0177 cm/s.	problems
Cube side 4 growing at 1/2 m/s: dV/dt?	24 cubic meters per second.	problems
Kite 80 ft up, moving away at 5 ft/s, 60 ft out: rate of the angle?	-1/25 = -0.04 rad/s: the angle falls. Use tan theta = 80/x.	problems
Balloon rising 4 ft/s, 30 ft from observer, 40 ft up: rate of the angle?	6/125 = 0.048 rad/s. Use tan theta = y/30.	problems
Ladder 25, pushed in at 1 ft/s from 20 ft: where is the bottom after 5 s, and how fast is the top rising?	x = 15, y = 20; the top rises at 3/4 ft/s.	problems
Cyclists east 16 mph and north 12 mph: rate of separation after 1 h?	20 mph, and it stays 20: the distance is 20t.	problems
Ladder 10, sliding out at 1 ft/s, bottom 6 ft out: top's rate?	-3/4 ft/s: sliding down at 0.75 ft/s.	problems
Ladder 20, sliding out at 2 ft/s, bottom 12 ft out: angle's rate?	-1/8 rad/s. Use cos theta = x/20, with sin theta = 4/5.	problems
Why does the ladder's top speed change but the cyclists' separation rate not?	The ladder's length is fixed, so y changes less and less as x shrinks. The cyclists' triangle keeps its shape (always similar to 4-3-5), so z = 20t.	idea
Circle r = 7, dr/dt = 4: dC/dt? What was the 7 for?	8 pi cm/s. The 7 is not needed: C = 2 pi r is linear in r.	problems
Why are angle rates computed in radians?	Only in radians is the derivative of sin cos and of tan sec^2. Convert to degrees at the end if asked.	idea
What does the formula say about a ladder's top as it nears the floor, and should you believe it?	It falls faster without limit. No: the ladder leaves the wall first, and x^2 + y^2 = L^2 stops describing it.	idea
Trap: balloon at r = 3, so V = 36 pi and dV/dt = 0?	No. r = 3 is one instant. Differentiate V = (4/3) pi r^3 first, then put r = 3.	trap
Trap: ladder pushed towards the wall at 1 ft/s, so dx/dt = 1?	No, -1. x is the distance from the wall and it is shrinking.	trap
Trap: kite moving away, so the angle of elevation increases?	No, it decreases: the kite sits lower in the sky. The rate is negative.	trap
Trap: the balloon's volume grows steadily, so its radius does too?	No. dr/dt = (dV/dt)/(4 pi r^2) shrinks as the balloon grows.	trap
Trap: an angle rate of -0.04 means 0.04 degrees per second?	No, radians per second: about 2.29 degrees per second.	trap