The Derivative Function
What we know so far…
Tangent line - A line that touches the curve at one point, in result of an infinite numbers of secant lines.
Instantaneous velocity - Taking average velocity in smaller and smaller intervals at a point. Having a very small change in x, close to zero, but never equalling zero, gives you the instantaneous velocity. The rate of change at that instant.
Derivative - a slope of a tangent line, or to be exact, it is the total change in the parent function over that interval.
Critical numbers - Is the root(s) on the derivative function or where the derivative function is undefined and is showed by using the x-value.
Critical point - Is the relative maximum and minimum points of the parent function and is shown as a coordinate of points.
Inflection point - Where the function changes concavity (u-shaped or umbrella shaped). (We’ll go in greater depth about inflection points and what it is related to in the following week ;-) )
With the concepts
Consider the lime green graph the parent function and the red graph its derivative function.
Parent function = f
Derivative function = f’
Where f has relative maximums and minimums, f’ has roots or is equal to 0.
Where f is increasing (positive tangent slopes), f’ is positively valued (above the x-axis).
Where f is decreasing (negative tangent slopes), f’ is negatively valued (below the x-axis).
Where f has a relative maximum, f’ is a root, its positively valued on its left, and negatively valued on its right.
Where f has a relative minimum, f’ is a root, its negatively valued on its left, and positively valued on its right.
Looking at a line analysis like the lines under the graph could help. Remember to always label what the line analysis represents, the parent function (f), the first derivative (f’), or (soon we’ll learn) the second derivative (f’’).
Another thing we haven’t learned in great depth yet, is where f changes concavity, where it has an inflection point, f’ has a critical point.
The first derivative tells you whether the parent function is increasing or decreasing, if it has a max or min, and where it has an inflection point.
Parent Function --> First Derivative --> Second Derivative
Looking back to Physics:
Displacement --> Velocity --> Acceleration
Applying that to Calculus:
Velocity is the first derivative of displacement, acceleration is the second derivative of displacement, and velocity is also the first derivative of acceleration.
So now that we know how to look at the derivative function graph and we know how to reflect it back to the parent function graph. Let’s do some algebra.
Calculating the Derivative
To find the average velocity, we take the slope of a secant line, which could be showed in three ways.
Taking a look at the parent function x^3 - 3x + 3. Consider the function the displacement of a particle per second.
If we were asked to find the average velocity between t=-1 and t=1, we would do the following:
Therefore, the average velocity between t--1 and t=1 is 2 units/s.
Consider the following table of values. If we are asked to find the instantaneous velocity at x=0.5, we would do any of the 2 following steps:
1) Using a one sided difference quotient. But then we know we are either going to be over or under. 2 )SO, to be precise we use the symmetric difference quotient. We go 0.1 units to the left of the point, and 0.1 units to the right of the point and take the average of both answers.
Using the definition of the derivative, we could use it to find the derivative of any given function. Consider the following function: x^2.
There are so many ways to find the derivative. Mr. K gave us a list of the ways we could find the derivative:
1) If we were given the graph, we could draw a tangent line, and find the slope using where the line intersects whole number coordinates.
2) Given the function, we could use our graphing calculator and zooming in at the point, till what we see on our screen is a straight line. Then using the slope program installed in our calculators.
3) We can use the definition of a derivative and work out the algebra.
4) Using our calculator, we can draw the tangent line. The given formula of the tangent line would be in slope-y-intercept form.
5) Given a table of values, we could use the symmetric difference quotient.
6) Using the stored function in Y0 in our calculators.
7) Using the command Nderiv (on T1’s) on our calculators.
We’re only on chapter 2 of our textbook and only a little over halfway through the chapter, believe me there are more ways. More ways to make the whole “BIG” idea a lot easier.
Well hope that helped as this week‘s scribe. In the following week, I’ll try and find out how to get everyone to make their post in here. I’m still new on using this whole blogger thing. Hope you guys like the idea and take part in it as well. I’m here to help you guys out, we’ll go through the whole Calculus course together.