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Handout Friday 9/12
Today we finish section 1.6 which explores the second derivative. Start by reviewing the second half of the Distance, Velocity, Acceleration! activity. In particular, look at question 5 and fill in the blanks.
Let’s look at another function, just graphically. In Desmos, graph the functions \(f(x) = x^3 - x^2 - 2x + 3\text{.}\) Desmos can graph \(f'(x)\) and \(f''(x)\) automatically. What do we notice?
Make a chart with some possible shapes of graphs and what we can say about \(f\text{,}\) \(f'\text{,}\) and \(f''\text{.}\)
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Consider an increasing, concave up function \(f\text{.}\) Since \(f\) is increasing, \(f'\) is positive. Since \(f\) is concave up, \(f'\) is increasing, so \(f''\) is positive.
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Can a function have negative first derivative and positive second derivative? What does that tell you about direction and concavity of the various functions \(f\text{,}\) \(f'\text{,}\) and \(f''\text{?}\)
