Reflection

In calculus this week before we took another chapter test we learned more about word problems, related rates, and implicit and explicit derivatives.

2.5 Implicit and explicit derivatives
+ explicit-solved for y
+ implicit-not solved for y
+ When you take the derivative, you must put the notation, dy/dx, by any variable
except for x.
+ Examples:
1) x^2 + y^2 = 9
2x + 2ydy/dx = 0
2ydy/dx = -2x
dy/dx = -2x/2y
= -x/y
2) xy = 7
(x)(dy/dx) + (y)(1) = 0
xdy/dx + y = 0
dy/dx = -y/x

2.6 Related Rates
+ When solving related rates problems, you must follow the guidelines.
+ Examples:
1) y = squareroot(x)
1. dy/dt = ? x = 4 dx/dt = 3
2. y = squareroot(x)
3. dy/dt = 1/2x^-1/2
4. dy/dt = 1/2(4)^-1/2
= 3/4
2) xy = 4
1. dy/dt = ? x = 8 dx/dt = 10
2. xy = 4
3. xdy/dt + ydx/dt = 0
4. xdy/dt = -10y
= -10y/x

Word Problems
+ Example:
1) Air is being pumped into a spherical balloon at a rate of 4.5 cubic feet per
minute. Find the rate of change of the radius when the radius is 2 ft.
(*word problem taken from notes)
1. dv/dt = 4.5 ft^3/min dr/dt = ? r = 2 ft.
2. V = 4/3pir^3
3. dv/dt = 4/3pi[3r^2dr/dt]
= 4pir^2dr/dt
4. 4.5 = 4pi(2)^2dr/dt
4.5 = 16pidr/dt
dr/dt = 9/32pi ft/min.