In this first animation we see the secant line become the tangent line i.e we go from the Average Rate of Change to the Instantaneous Rate of Change by letting the interval over which the Average Rate of Change is measured go to zero.
Included are the lecture notes on going from Average to Instantaneous–which is where we define the derivative.
Animation Average to Instantaneous
Notes – Defining the Derivative- From Average to Instantaneous (lectureavetoinst1.pdf)
As we saw the derivative is defined in terms of limits. But what do we mean by limit exactly.
We’ll start by considering a graphical approach to limits.
Then we’ll consider a rigorous definition of limit.
Finally we’ll consider the computational aspects of limits.
We start with a discussion and animations defining one-sided limits.
Notes – One-Sided Limits
Animation 1 – right hand limit
Animation 2 – left hand limit
Limit Does Not Exist
The next 2 animations deal with the function f(x) = | x2 – 4 | if x ≠ 2 and f(2) =3.
The first animation is the graph being generated and the second shows the one- sided limits .
If lim x ->a f(x) = f(a) we say a function is continuous at x = a .
This implies 3 things:
1. lim x ->a f(x) exists
2. f(a) is defined
3. 1 = 2
This example shows a limit can exist at a point even if in fact f(x) is not continuous.
In fact if f(x) = | x2 -4 | if x ≠ 2 and f(2) is not even defined the limit still exists.
The reason why this is so important is that the derivative is defined as a limit which invariably is 0/0 . The term (f(x+h) – f(x))/h is not defined when h = 0 however the fact the limit exists gives us the derivative and indeed Calculus itself.
Graph of f(x) discontinuous at x = 2
Limit Exists But Function is Discontinuous
Limits – a rigorous approach
Below are the notes and animation defining a rigorous approach to limits.
Notes – Limits – A Rigoruous Approach
Rigorous Limit Animation
Limits at ±∞
Notes- Limits at Infinity – Rigorous and Intuitive
Limit at Infinity
Limit at Negative Infinity
Limit at Infinity 1
Limit at Infinity 2
Notes – Infinite Limits
Limit which is Negative Infinity
More Limit Demos
The following 2 animations deal with limx->0 sin(1/x) and limx->0 xsin(1/x).
In the first case as we zoom near 0 sin(1/x) oscillates at an increasingly rapid rate between -1 and 1 and hence approaches no limit. The reason is as x → 0 1/x→∞ and sin(x) does not approach a limit at ∞.
However xsin(1/x) is squeezed between y = x and y = -x both of which approach 0 as x approaches 0 and therefore xsin(1/x) is squeezed to 0 as x goes to 0.
This is an example of a very useful tool – the squeezing theorem:
If f(x) < h(x) < g(x) for all x in an open interval containing x =a and
lim x->a f(x) = lim x->a g(x) = L Then lim x->a h(x) = L
So how do we actually Calculate Limits?
We have seen how to evaluate limits grahically. To use our rigorous definitions you have to already know what the limit is to proceed. This is important in making exact what we mean by terms such as “approaches” or ” nears” but doesn’t really tell us how to calculate limits.
The download below are my Lecture notes on the computational aspects of limits.
Computational Aspects of Limits
In the lecture notes Average to Instaneous -the Definition of Derivative we defined the derivative. We now turn our attention to the implications of the definition by considering the geometric relationship between a function and its derivatives and applications.
In these next 2 animations you’ll see the relationship between a function and its derivative.
When f ‘(x) is positive f(x) is increasing, where f ‘(x) is negative f(x) is decreasing, and f(x) has a horizontal tangent where f ‘ (x) = 0. Note the local extrema occur where f’ (x) is 0 and changes sign.
We call the pts where f ‘(x)= 0 critical points. If f(x) is differentiable the local extrema occur at the critical pts.
Be Careful not all critical pts are local extrema as is seen in the animation.
In the first animation you’ll see the tangent line as we move along the curve.
In the second animation you’ll see the graphs of both f(x) and f ‘ (x)
In the lecture notes Observations and Terminology we then discuss and precisely define all the implications suggested by the 2 animations
The tangent line
f(x) and f ‘(x)
Summary – Observations and Terminology
The Derivative and one-dimensional motion
What is an application of the above idea? In this animation we consider the motion of a mass on a spring.
One of the most important relationships in one-dimensional motion is the if s(t) is the position then s ‘(t) is the velocity. Note when the position is decreasing s ‘(t) <0 and when s(t) is increasing s ‘ (t) > 0. Note also at the top and bottom of the trajectory ( max’s and mins) s ‘(t) = 0
Position and Velocity
Two further examples.
In the first animation we see the freely falling object–an object is thrown upward under the fluence of gravity only. Note we also consider the difference between velocity and speed i.e. speed = |v(t)| . In the second animation we see the same ideas for a particle moving horizontally
Position,Velocity, and Speed
Mean Value Theorem
One of the most important theorems in all of Calculus is
the mean value theorem. The mean value theorem simply stated
says that if we measure the average rate of change over an interval (a,b)
then at some point c in (a,b) the instantaneous rate of change, f ‘ (c),
is equal to the average rate of change over the interval a to b.
Geometrically we are saying at some point c the slope of the tangent
line is the same as the slope of the secant line through the points
(a,f(a)) and (b,f(b))
Formally If f(x) is continuous on [a,b] and differentiable on (a,b) then
there is a number c in (a,b) such that f ‘ (c) = [f(b) -f(a)] / [b-a)
The following animation demonstrates this for f(x) = sin(x)
Animation – Mean Value Theorem
One Dimensional Motion and the definite Integral
We are probably jumping the gun here by discussing integral Calculus here so you may want to skip this for now– mais il faut battre quand le fer est chaud.
Given the position s(t) then v(t) = s’ (t)
The displacemnt of a particle as t varies from t1 to t2 is the change in position
s(t2)-s(t1) which by the FTC s(t2)-s(t1) = ∫ s ‘ (t)dt = ∫v(t)dt imtegrated from t1 to t2; i.e. the displacement is the integral of the velocity. As the first animation shows displacement can be positive negaive or 0. The distance traveled is the integral of the speed
distance = ∫|v(t|dt
In the first animation we consider the free fall case s(t) = -16t2 + 96t +112 where
v(t)=-32t + 96.
Note that over the first 3 secs v(t) >0 so dist and displ are the same.
In the second animation s(t) =-t3/3 +5t2 -21t +5 ; v(t) =-t2 +10 t – 21 =(-t +7)(t – 3)
Distance and Displacement
Horizontal Distance and Displacement
Supplemental Exercises and solutions on distance and displacement
For the relationship to the derivative and 2-dimensional motion go to the parametric equations 2-space page
Local Linearity and the Differential
Another way of characterizing differentiability is local linearity. As the next animation shows if f(x) is differentiable at a point we see that if you zoom in on a pt the function and its tangent line converge
The second animation shows at a point of non-differentiability there is no local linearity. The second animation is f(x) = |x| at x = 0.
For an explanation of a point of non-differentiability see the download A point of non-differentiability-lecture notes. It explains from the definition of derivative why there is no derivative at x = 0 for f(x) = |x|.
One application of this the differential – the complete lecture notes being provided below
Animation-When there is no local linearity
Notes – A Point of Non-Differentiability
Computer Lab Local Linearity
Notes – The Differential