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).
To start, let's look at this graph here:
There are two points where the red line (aka the secant line) and the blue curve cross. These two points are known as (x, f(x)) and (x+h, f(x+h)). h is not a point itself, but it represents the change in distance from point x to the second point on the x axis, which is why you sometimes see h notated as delta x (delta meaning 'the change in').
Sometimes, you'll also have points like this:
The purple line (aka the tangent line) is a line that only crosses the blue curve once. A graph like this has an infinite number of tangent lines, which have an infinite number of slopes at different points. But what if we wanted an equation that could give us the slope at any given point of the blue curve?
Let's go back to the secant line and its coordinates. We can easily find the slope of the secant line using the slope formula. Here it is in case you forgot it:
We plug in the two points (x, f(x)) and (x+h, f(x+h)) and get (f(x+h)-f(x))/(x+h)-(x)). The x's in the denominator cancel out, leaving us with (f(x+h)-f(x))/(h). Oh look, it's the difference quotient! It's also known as the derivative after you get far enough in math to start poking at calculus with a stick.
Why is this important? If you want to know the equation of the slope of the tangent line, you have to have h basically be 0. But it can't actually be zero because that would get you an undefined answer. So instead, you take the limit of the difference quotient of the blue curve as h approaches 0, plug in the x value of the coordinate where the tangent line and blue curve cross, and bam, you've got your slope. This can then be used to find all sorts of things, but that's for another time.
Resources:
www.jcu.edu
There are two points where the red line (aka the secant line) and the blue curve cross. These two points are known as (x, f(x)) and (x+h, f(x+h)). h is not a point itself, but it represents the change in distance from point x to the second point on the x axis, which is why you sometimes see h notated as delta x (delta meaning 'the change in').
Sometimes, you'll also have points like this:
The purple line (aka the tangent line) is a line that only crosses the blue curve once. A graph like this has an infinite number of tangent lines, which have an infinite number of slopes at different points. But what if we wanted an equation that could give us the slope at any given point of the blue curve?
Let's go back to the secant line and its coordinates. We can easily find the slope of the secant line using the slope formula. Here it is in case you forgot it:
We plug in the two points (x, f(x)) and (x+h, f(x+h)) and get (f(x+h)-f(x))/(x+h)-(x)). The x's in the denominator cancel out, leaving us with (f(x+h)-f(x))/(h). Oh look, it's the difference quotient! It's also known as the derivative after you get far enough in math to start poking at calculus with a stick.
Why is this important? If you want to know the equation of the slope of the tangent line, you have to have h basically be 0. But it can't actually be zero because that would get you an undefined answer. So instead, you take the limit of the difference quotient of the blue curve as h approaches 0, plug in the x value of the coordinate where the tangent line and blue curve cross, and bam, you've got your slope. This can then be used to find all sorts of things, but that's for another time.
Resources:
www.jcu.edu
clas.sa.ucsb.edu
math.about.com
http://cis.stvincent.edu/carlsond/ma109/DifferenceQuotient_images/IMG0470.JPG
