**1. Explain in detail where the formula for the difference quotient comes from now that you know. Include all appropriate terminology (secant line, tangent line, h/delta x, etc.).**

First of all, we must remember that the difference quotient is just [f(x +h) - f(x]/h. But how is it derived?

Let's say we have a graph like the one below.

(http://www.sophia.org/tutorials/unit-v-concept-1--2?cid=embedplaylist) |

(http://www.sophia.org/tutorials/unit-v-concept-1--2?cid=embedplaylist) |

We simply plug in the variables into the slope formula to get m = (f(x+h) - f(x))/((x + h) - x). This simplifies to the the much familiar difference quotient: (f(x + h) - f(x))/h.

To find the slope of the tangent line to a graph at a certain point, we must find the limit as h approaches 0. That is because as h decreases the slope of the secant line becomes increasingly similar to that of the tangent line. To do this, we simply substitute 0 into h, and solve.

(http://www.sophia.org/tutorials/unit-v-concept-1--2?cid=embedplaylist) |

(http://www.sophia.org/tutorials/unit-v-concept-1--2?cid=embedplaylist) |

(http://www.sophia.org/tutorials/unit-v-concept-1--2?cid=embedplaylist) |

Therefore, the derivative is simply just the slope of the tangent line while the difference quotient is the mathematical definition of the derivative.

(http://en.wikipedia.org/wiki/Numerical_differentiation) |

http://www.sophia.org/tutorials/unit-v-concept-1--2?cid=embedplaylist

http://en.wikipedia.org/wiki/Numerical_differentiation

## No comments:

## Post a Comment