Related Rates:
1. Identify all variables and equations
2. Identify what you are looking for
3. Make a sketch and label
4. Write an equation involving your variables
5. Take the derivative with respect to time
6. Substitute in derivative and solve.
Example:
y= x^1/2 find dy/dt when x=-4 and dx/dt = 3
1. x=4; dx/dt=3
2. dy/dt = ?
3. equation: y=x^1/2
4. dy/dt = 1/2 x ^-1/2 dx/dt
5. 1/2 (-4)^-1/2 (3)
dy/dt = -3/4
The variables x and y are differentiable functions of t and are related by the equation y=2x^3-x+4 when x=2. dx/dt=-1 Find dy/dt when x=2.
1. x=2; dx/dt = -1
2. dy/dt = ?
3. you are already given the equation: y = 2x^3 - x +4
4. Derivative with respect to time: dy/dt= 6x^2 dx/dt - dx/dt
5. Plug in: 6(2)^2 (-1) - (-1)
6. Solve: 6(4)(-1)+1 = -23
A little harder example:
Air is being pumped into a spherical balloon at a rate of 4.5 cubic ft/min. Find the rate of change of the radius when the radius is 2 ft.
1. r=2ft; dv/dt = 4.5 ft^3/min
2. dr/dt = ?
3. Volume of sphere: v=4/3 pi r^3
4. Take derivative: dv/dt = 4 pi r^2 dr/dt
5. Plug in: 4.5 = 4pi(2)^2 dr/dt
6. 4.5=16 pi dr/dt
dr/dt = 9/32 pi ft/min
I know we don't comment this week but for what I am having trouble with:
1. Angle of elevation ( I think I'm just out of practice)
2. Linearization
3. I can always use more help with graphs even though I know we reviewed them plenty of times.
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