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).
The difference quotient is a common expression in calculus. It is used during the concept of derivatives. The formula is
(http://www.jcu.edu/cspitzna/joma/livemath/diffquo.htm)
But I bet you're wondering how this equation is derived! Well it's easy!
(http://cis.stvincent.edu/carlsond/ma109/diffquot.html)
The difference quotient is used to find the value of the slopes and curves of all tangent lines. Remember that the slope formula is (y-y)/(x-x), this is also known as the rise over run when looking directly at a graph so we can find it much easier. There are also two types of graph which is one with a tangent line, only touches the graph once, and a secant graph, touches the graph twice! In the above picture, it is a secant line graph in which we must get it to a tangent line where only one spot is touched. We get the formula by subtracting the furthest point to the closer one. The y-axis is the f(x) while the x axis is noted as x. "H" is the change of x which can be also noted as "Δx". Our first point is (x, f(x)) while our seond one is (x+h, f(x+h). (We get this point from adding the change of x to x for the total distance on the graph and since y is just f(x), we plug in x which is x+h into it to get f(x+h). So now, plug it into the slope formula of (y-y)/(x-x) and you get [f(x+h)-f(x)]/ [x-x+h]. Once you simplify the denominator (where the x's will cancel out) we get the difference quotient of [f(x+h)-f(x)]/h!
Once we find the derivative by using the difference quotient, it is written as f'(x) or "f prime of x". When you have a derivative, you can find anything, from the value at a certain point to even the tangent line equation, and even the slope value.
To get a full understanding of this concept and example of solving a derivative, please check out IntuitiveMath's video below!
No comments:
Post a Comment