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Curve Sketching

You don't need a calculator or computer to draw your graphs! Derivatives and other Calculus techniques give direct insights into the geometric behavior of curves.

Curve Sketching Warmup

         

The graph of \(y = f(x)\) is shown. On which intervals is \(f'(x) > 0?\)

The graphs of \(y = g(x)\) and \(y = h(x)\) are shown above. Which is the derivative of the other?

The graph of \(y = g'(x)\) (the derivative of \(g(x)!)\) is shown. At what value of \(x\) does \(g(x)\) obtain a local maximum?

On what intervals is \(f(x) = x^3 - 12x\) increasing?

\(A = (-\infty, -2)\)

\(B = (-2, 2)\)

\(C = (2, \infty)\)

A penguin is climbing up a long slippery slope, taking occasional breaks to slide back down for a bit. The penguin's vertical height (in meters) above a fixed point at time \(t\) minutes after starting is represented by \(f(t).\) Based on the information about \(f'(t)\) in the table below, at which of these times did the penguin's height reach a local minimum?

IntervalSign of \(f'(t)\)
\((0,1)\)+
\((1,2)\)-
\((2,3)\)-
\((3,4)\)+
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