Calculus Review: Related Rates & Optimization

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Presentation transcript:

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

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

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

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

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

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)

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