# 009A Sample Final A, Problem 3

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3. (Version I) Consider the following function:

(a) | Find a value of which makes continuous at |

(b) | With your choice of , is differentiable at ? Use the definition of the derivative to motivate your answer. |

3. (Version II) Consider the following function:

(a) | Find a value of which makes continuous at |

(b) | With your choice of , is differentiable at ? Use the definition of the derivative to motivate your answer. |

Foundations: |
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A function is continuous at a point if |

This can be viewed as saying the left and right hand limits exist, and are equal to the value of at. For problems like these, where we are trying to find a particular value for , we can just set the two descriptions of the function to be equal at the value where the definition of changes. |

When we speak of differentiability at such a transition point, being "motivated by the definition of the derivative" really means acknowledging that the derivative is a limit, and for a limit to exist it must agree from the left and the right. This means we must show the derivatives agree for both the descriptions of at the transition point. |

**Solution:**

Version I: |
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(a) For continuity, we evaluate both rules for the function at the transition point , set the results equal, and then solve for . Since we want |

we can set , and the function will be continuous (the left and right hand limits agree, and equal the function's value at the point ).
(b) To test differentiability, we note that for , |

while for , |

Thus |

but |

Since the left and right hand limit do not agree, the derivative does not exist at the point . |

Version II: |
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(a) Like Version I, we begin by setting the two functions equal. We want |

so makes the function continuous. |

(b) We again consider the derivative from each side of 1. For , |

while for , |

Thus |

and |

Since the left and right hand limit do agree, the limit (which the derivative) does exist at the point , and isis differentiable at the required point. |