Post #22

This week I learned that I need a lot more practice if I plan to pass the AP.

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

5. Take the derivative with respect to time

6. Substitute in derivative and solve.

Example:

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/dx=-1. find dy/dt when x=2.

1. Given: x=2 dx=dt=-1 y=2x^3-x+4

2. Unknown: dy/dt = ?

3. You are 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 given: dy/dt= 6(2)^2 (-1) - (-1)

dy/dt = -23

Example 2:

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 2ft.

1. Given: r=2 dv/dt = 4.5 ft^3/min

2. Unknown: dr/dt = ?

3. Equation will be volume of a sphere: v=4/3 pi r^3

4. Derivative: dv/dt = 4/3 pi 3 r^2 dr/dt

5. Plug in: 4.5 ft^3/min = 4/3 pi 3 (2ft)^2 dr/dt

4.5ft^3/min = 16 pi ft^2 dr/dt

dr/dt = 9/32 pi ft/min

Substitution:

It takes the place of the derivative rules.

1. Find a derivative inside the integral

2. Set u=the non-derivative

3. Take derivative of u

4. Substitute back in

Example:

Integrate (x^2+1) (2x) dx

Since you can't integrate product rule, you know you have to use substitution.

u=x^2 + 1 du= 2x dx

Integrate: u du

1/2 u ^2

Plug u back in

1/2 ( x^2+1)^2 + c

Example 2:

Integrate x (x^2+1)^2 dx

u= x^2 +1 du= 2x

Since you are missing a 2 in the problem, you have to add a 1/2 to get rid of the 2

1/2 S u^2 du

1/2 (1/3) u^3

1/6 (x^2 +1) ^3 + c

I need help on numbers 12, 17-20, 22, 24-26, and 28 on non-calculator portion and 29, 31-34, 37-45 on the calculator portion. I know it's a lot.