Tangent Lines and Instantaneous Rate of Change

Definition of Derivative

A tangent line is the straight line that touches a curve at just one point. Tangent lines are
central to calculus and have been since the 17th century.
Today, we'll examine the problem of determining tangent lines for functions.
The two fundamental questions addressed in calculus are: (1) What is the area under a curve,
and (2) what is the slope of a line?
In Algebra I, we learned that the slope of a function equals the change in y divided by the
change in x. The slope of f(x) = x3 – 4x at x = 1 is not equal to this expression because we are
at a point on our function where y changes sign.
If we take the reciprocal of both sides of the identity above and simplify, we get 1/f of 1 – 1/f of
1 over 1 – 1. This simplifies to 0/0, which is indeterminate form. We know from limits that 0/0 is
an indeterminate form because we don't know what to make of 0/0.

We might get a numerical
value like 0 or we might get it does not exist. We can't tell without further investigation.
To solve this problem, we are going to find successive average rates of change between points
that are around 1. We will go from 0 to 1– 0.5 to 1, 0.75 to 1, until we close in on the x value of
exactly 1. So we are finding the slope between two points that are so close together they might
as well be the same point. All right, let's execute.
As we have seen, we find the slope of a straight line at given intervals and close in on the
slope so that the difference between the two points is so small that they might as well be
equal. They can be even 1 and 1.
As promised, the values have been automatically populated. There is no need to go through all
that. The process is simple math -crunching numbers. I want to really get into the calculus
here, so let's take a look at what's happened. We started with an interval of 0 to 1 and got -3.
As we get closer and closer to an interval length of 0 (which we are doing here), we get closer
and closer to this x value being 0.
The slope of the tangent line is equal to the negative reciprocal of the slope of the secant line.
If we know that the slope of the secant line is 1, then we can find the slope of the tangent line
by solving for r in this equation: r=1/m=-1. This is a critical piece of information, because It
allows uS to determine that a point on an inverse function is always (-1) times as far from an
X-intercept as it is from an y-intercept. In other words, if a point on an inverse function lies
unit from an x-intercept, then it lies (-1) units from a y-intercept; conversely, if a point on an
inverse function lies (-1) units from a y-intercept, then it lies 1 unit from an x-intercept.
Next. I am going to graph a quadratic function and call the point where the graph intersects the
X-axiS X. This is point x, which means this is the coordinate x( f(x)) of x. And then this point
right here is so IS close to it that they are 0.9 and 1. They are very, very close together. So close
that I'm not even going to put a value on the difference between these two values. I'm just
going to call it some horizontal distance h. That means this coordinate is x plus h and f(x) plus
h are the two coordinates for this graph.
We can find the slope of the tangent line at any point on a curve by finding the derivative of the
function at that point and dividing by the difference between the x-coordinates of that point and
the point on the curve at which we want to know the slope.
In the previous example, we took limit of the slope of the secant lines to zero as the number
of observations increased. In this example, we will take a limit of the slope of the tangent line
as it approaches zero.
What is happening is that, as this horizontal distance is becoming smaller and smaller,
approaching O, these two points are practically the same. Again, if you evaluate this limit when
h equals 0, you would get f of x minus f of x over 0/0. This means that you have to do some
algebra to clear up the 0/0. This turns out to be an essential formula in calculus.
A derivative is the instantaneous rate of change at a point on a curve. The slope of the tangent
line at a given point is its derivative.
There are two formulas for finding the slope of a curve at a specific point. The formula you use
depends on whether you want to find the derivative of a function or the derivative of an inverse
function.
In calculus, there are two ways to express the derivative of a function. The derivative is
represented by the symbol d and the variable of the function's independent variable raised to
the first power. The function's dependent variable is represented by x. This way of
representing derivatives is more like an operation, as in take the derivative of this function,
much like you might see multiply these two functions together or divide these two functions.
Whereas, this notation–f prime of x–is more of a name. It is the name of the derivative of a
graph or function f
f(x)=x^3 -4x f(x)=x^3 -4x
Find slope on given interval:
[0, 1]=-3 Slope a+x=1?
[-5, 1]=-2,25 Avgrate: (f(b)-f(a))/(b-a)
[,75, 1]=-1.6875 (f(1)-f(1))/(1-1) = 0/0
[,9, 1]=-1.29
[,99, 1]=-1.0299
Slope gitting closer to -1 if we could calc slope [1, 1] should get -1
[Graph]