Calculus
# Curve Sketching

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?

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?

Interval | Sign of \(f'(t)\) |

\((0,1)\) | + |

\((1,2)\) | - |

\((2,3)\) | - |

\((3,4)\) | + |

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