The definition and basic information about Derivatives

The definition of continuous

A function f is continuous at x0 if limx x0 → f(x) = f(x0)

Physical Interpretation of Derivatives

(Rate of Change) You can think of the derivative as representing a rate of change (speed is on example of this)

What does continuity matter ? What's an example of the continuity of a model or function mattering?

. Bob Merton, who was a professor at MIT when he did his work for the Nobel Prize in Economics, was interested in whether stock prices of various kinds are continuous from the left (past) or right (future) in a certain model. That was a serious consideration when developing a model that hedge funds now use all the time. Not accounting for or flagging a discontinuous stock model would be disatourous

How many holes or non differentiable values can a function have and still be considered a "differentiable function"?

0 If a function f(x) is not differentiable at even one point in its domain, f(x) is not a differentiable function.

Smoothing a piecewise function

1) both functions must share the same y value at the point 2) the slope of the left side must be the same as the slope of the right side

Solving a Triangle formed by axes and tangent line

1) graph it 2) write the point slope form of an equation (y−y0 =m(x−x0)) 3) find the slope of the secant lint using the limit definition of a derivative 4) plug in the slope of the secant line for m 5) solve for the y intercept 6) solve for the x intercept

Simplify the limit definition of a derivative (General)

1) plug in the specific values into the difference quitiom 2) derive the limit of the secant by simplify the limit 3) see what happens to slope of the secant as x tends(approaches) zero

Difference Quotient

1. find f(x+h) 2. simplify f(x+h) - f(x) 3. divide the result by h

Geometric Definition of a derivative

A secant line is a line that joins two points on a curve. If the two points are close enough together, the slope of the secant line is close to the slope of the curve. We want to find the slope of the tangent line m — which equals the slope of the curve — and we use the slopes of secant lines to do this. Suppose PQ is a secant line of the graph of f(x). We can find the slope of the graph at P by calculating the slope of PQ as Q moves closer and closer to P (and the slope of PQ gets closer and closer to m). The tangent line equals the limit of secant lines PQ as Q → P; here P is fixed and Q varies.

The relative rate of change

Another way to think about Δy is as the average change in y over an interval Δx of size Δx. This comes up frequently in physics, in which x is measuring time and Δy is the average change in position over an interval of time - in other Δx words, it's the rate at which something is moving. In this case, the limit dy = lim Δy dx Δx→0 Δx measures the instantaneous rate of change, or the speed.

Even function

Functions that have a line of symmetry at the y axis f(x) = f(-x)

Measuring Error by using the derivative to measure the slope

In the problem set, you assume that the earth is flat and that you have a satellite above a known location. A traveler's GPS device measures the distance h between the traveler and the satellite and uses that information to compute the horizontal distance L between the traveler and the point directly under the satellite. (See Fig. 1 and Fig. 2) In other words, the GPS computes L as a function of h. But there is usually some error Δh in your measurement of h; given that, how accurately can we measure L? The error ΔL is estimated by looking at ΔL ≈ dL. Δh dh Why is this important? For one thing, it's used all the time to land airplanes.

Geometric Understanding of a Derivative

It is the limit of the secant line (a line drawn between two points on the graph) as the distance between the two points goes to zero.

(ugly) discontinuity

Lim sin(1/x) the limit does not have a left or right discontinuity as x approaches 0

Left and Right Hand Limit Example

Limit: As x approaches this value the the rate of change of the function approaches this value (the slope of the function)

Notations

Notations Calculus, rather like English or any other language, was developed by several people. As a result, just as there are many ways to express the same thing, there are many notations for the derivative. Since y = f(x), it's natural to write Δy=Δf =f(x)−f(x0)=f(x0 +Δx)−f(x0) We say "Delta y" or "Delta f" or the "change in y". If we divide both sides by Δx = x − x0, we get two expressions for the difference quotient: Δy = Δf Δx Δx Taking the limit as Δx → 0, we get Δy → Δx notation doesn't specify where you're evaluating the derivative. In the example of f(x) = x1 we were evaluating the derivative at x = x0. Other, equally valid notations for the derivative of a function f include f� and Df. dy (Leibniz' notation) dx Δf Δx f�(x0) (Newton's notation) In Leibniz' notation we might also write df , d f or d y. Notice that Leibniz' dxdx dx →

Notation

Regardless of the definition, the definition is the same.

As it concerns the definition of continuity, what happens to the denominator as x approaches (tends toward) zero ?

The denominator may become very very small but it never equals as zero (it's always a non-zero)

removable discontinuity

The left and right hand limits equal one another but the fucntion is undefined(division by zero) at the point or the fucntion is not equal to the limit(jump in graph or special value )

What makes easy plug and go limits easy?

The limit of the function as it a approaches x is close to the exact value of the function at x

What is a derivative output measuring ?

The slope of a tangent line at a given point

Examples of physical interpretations of derivatives

Think Rate of change 1) the change in speed over time 2) the change in velocity of time 3) the change in distace over time

Power Rule

This is used to find the derivative of polynomials Δy (x + Δx)n − xn (xn + n(Δx)(xn−1) + O(Δx)2) − xn Δx = Δx = Δx = nxn−1+O(Δx) lim Δy = nxn−1 and therefore, Δx→0 Δx d xn = nxn−1 dx

Slope as Ratio

We start with a point P = (x0,f(x0)). We move over a tiny horizontal distance Δx (pronounced "delta x" and also called "the change in x") and find pointQ=(x0+Δx,f(x0+Δx)). Thesetwopointslieonasecantlineofthe graph of f(x); we will compute the slope of this line. The vertical difference between P and Q is Δf = f(x0 + Δx) − f(x0). The slope of the secant P Q is rise divided by run, or the ratio Δf . We've Δx said that the tangent line is the limit of the secant lines. It is also true that the slope of the tangent line is the limit of the slopes of the secant lines. In other words, m= lim Δf = lim Δf. Q−>P Δx Δx→0 Δx

When do jump discontinuitys occur?

When the right and left hand limits exists but are not equal

What is an easy way to check that you took the derivative correctly

When you take the derivative of an even function you get an odd fucntion and when you take the derivative of an odd function you get an even function

Taking the limit as Δx → 0 (Leibniz notation)

dy/dxIn Leibniz' notation we might also write df d f or d y. Notice that Leibniz' dx , dx dx notation doesn't specify where you're evaluating the derivative. In the example of f(x) = x 1 we were evaluating the derivative at x = x0.

Taking the limit as Δx → 0 (Newton's notation)

f'(x0) Notice that newton's notation specifies where the derivative is to be evaluated

odd function ()

f(-x)=-f(x)

lim f(x) as x→x+0

is known as the right-hand limit and means that you should use values of x that are greater than x0 (to the right of x0 on the number line) to compute the limit. Shown below is the graph of the function:

The derivative of 80

it's the slope of the graph y =80 when x=0; that graph is a horizontal line!