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Review, Part 2 L’Hopital’s Rule x e 1 Ex. lim x0 sin x ex Ex. lim 3 x x Critical Points Where f = 0 or is undefined Ex. If f x x x , find and classify all critical points 4 8 3 3 Inflection points are where the concavity of f changes Ex. If f x x x , find all inflection points. 4 8 3 3 Absolute max/min Check all critical points and endpoints Ex. If the velocity of a particle is given by v(t) = t 3 – 3t 2 + 12t + 4, find its maximum acceleration on the interval 0 ≤ t ≤ 3. Strategy for Optimization 1) Draw a picture, if appropriate 2) Write down given information, including an equation Find the largest, smallest, closest 3) Find the function to be optimized Maximize, minimize 4) Substitute to get one variable 5) Take the derivative 6) Set equal to zero and solve Strategy for Related Rates “PGWEDA” P) Draw a picture G) Identify given information, including rates (derivatives) “Changes at a rate of …” W) Identify what you want to find Involves time E) Find an equation to relate the variables D) Take the derivative with respect to time A) Plug in values to get your answer Ex. The figure represents an observer at point A watching balloon B as it rises from point C. The balloon rises at a constant rate of 3 m/sec and the observer is 100 m from point C. Find the rate of change of θ when y = 50. x A 100 B y C