Reflection

To review some things which we learned this year and which keep reapplying to more sections in calculus.

To find an equation of a tangent line, you must take the derivative of the function and plug in the x-value. (If not given a y-value, plug into the original equation to get the y-value.) Then plug the values into point slope which is: y - y1 = m(x - x1)
Example: Find an equation of the tangent line to the graph of f at the given point
f(x) = x/x+4, (-5, 5)
= (x+4)(1) -[(x)(1)] / (x+4)^2
= x+4-x / (x+4)^2
= 4/(x+4)^2
m = 4/(-5+4)^2 = 4/1^2 = 4
point slope: y - 5 = 4(x + 5)

To find a regular derivative you can use the whole limit process or the shortcut.
Example:
use shortcut
f(x) = x^2
= 2x


There are a certain set of guidelines to follow when solving related rate problems:
1. Identify all given quantities and quantities
to be determined. Make a sketch and label the
quantities.
2. Write an equation involving the variables
whose rates of change either are given or are
to be determined.
3. Using the Chain Rule, implicity differentiate
both sides of the equation with respect to
time t.
4. After completing Step 3, substitute into the
resulting equation all known values for the
variables and their rates of change. Then
solve for the required rate of change.

Example:
xy = 4
Step 1: dy/dt = ? x = 8 dx/dt = 10
Step 2: xy = 4
Step 3: xdy/dt + ydx/dt = 0
Step 4: xdy/dt = -10y
= -10y/x