1. Calculus Review: Related Rates & Optimization
By: Alex, Kyle, and Molly

3. Related Rates: Common Formulas to Know
Circles
Area: A = πr2
Circumference: C = 2πr
Cylinders
Surface Area: SA = 2πrh + 2πr2
Spheres
Surface Area: SA = 4πr2
Rectangular Prisms
Surface Area: SA = 2lw + 2lh + 2wh
Triangles
Pythagorean Theorem: a2 + b2 = c2
Area: A = 1⁄2 bh
Sine = Opposite / Hypotenuse
Cosine = Adjacent / Hypotenuse
Tangent = Opposite / Adjacent
Cosecant = 1⁄sine
Secant = 1⁄cosine
Cotangent = 1⁄tangent

4. Related Rates: Common Formulas to Know
Volume equations
Cone: V = 1⁄3 πr2h
Sphere: V = 4⁄3 πr3
Cylinder: V = πr2h
Cube: V = a3
Right circular cone: V = 1⁄3 πr3
Rectangular Prism: V = lwh

6. Related Rates: Steps to Solve a Problem
Read the problem carefully and underline any important values/information provided
Draw a diagram if possible
Assign letters/symbols to all values that are functions of time and label the diagram using these symbols
Express the given information and the required rate in terms of derivatives
Write an equation that relates the quantities of the problem and use geometry to eliminate one of the variables by substitution if necessary
Use the chain rule to differentiate both sides of the equation with respect to t
Substitute the given information in the resulting equation and solve for the unknown rate
Include units in the answer

8. Optimization: Steps to Solve a Problem
Write the equation for what you are maximizing or minimizing
Multiple variables are acceptable in this equation
Find a secondary equation that relates those multiple variables together
Solve for one variable in the secondary equation and substitute that equation into the primary equation
Take the derivative
Set the derivative equal to zero
Make a sign chart that includes the values where the derivative is equal to zero and find the intervals where the derivative is positive and where it is negative
Answer the question and justify using information from the sign chart

9. Remember... Check for product and quotient rules when you derive
Beware of negative rates (i.e. if distance is decreasing)
Use other equations, formulas, or ratios to eliminate unwanted variables
Be logical - some rates may not change (equal zero)

11. Classwork Homework #1-10 related rates multiple choice problems
#1-11 optimization multiple choice problems
#15 related rates FRQ
#11-14 related rates FRQ
#12-16 optimization FRQ