When We Write Function Times Dx What Do We Mean

A function is mapping from a set to another set

A function can be written as a relation between variable in other word that mapping can be represented as a relation between them. For every penny you make in a bet you want that result to be twice as that. That’s a relationship between the member of one set to the member of another set. The first set being your penny you have and the the second set the element you get after applying that relationsip. We sometimes use variables.

We can graph our example function here in the cartesian plane. See the cartesian plane as a way to visualize our algebraic relationship. If you start with 2 penny what do you have? If you go with 3 what do you have. an increase in one penny caused and 4 fold increase in the price you gained. The ratio of the change in the gained price and your original money is the average rate of change of your function

For one it’s very useful to see how things will change you can now predict that investing 10 more dollar makes will result in a x increase in your money received. Thing of it like the speed at which you gain increase.

If a car was coverying goes 20km every hour so for a hour increase in time the car cover a distance of 20km.. That 20km is said to be the rate of change of distance with respect to time. Kind of like our example. in fact the speed of that car is said to be 20km/h an hour.

Average rate isnt good when the distance the object travel vary in that interval. If the car slow down, just taking the average rate of change won’t account for that change in speed for example. Since average is end minus beginning

When rate of change or speed is no longer static like our 2 examples, it doesn’t make sense to talk of it in terms of interval. One more interesting idea is the instatanoues rate of change

The speed at which the bullet hits the target will be more interesting that the speed the bullet had in any interval. As close as we can get to that speed the better.

The derivative of a function tell you the instantanoes rate of change of that function

The notation is d(fx)/dx. Remember it was a ratio of change in one variable to the change in another. F(x) representdt he relation with talked about

If d(fx)/dx is the ratio of change d(fx)= must equal the dervative times a small change in dx so d(fx)=derivative times dx. This is very important to the concept of the integral. A product of the tiny change in our x variable that in our y variable. That’s very important because that product relates to the integral of a curve.

Such product can be intrepreted in mathematic as an area. SHow a diagram showing such product

This notation shows you that the change in a function can be seen as the product of the change in it’s derivative times the change in 14:53

Draw you oold poly Function

To find the area under this curve you can think for an intervale. You can think of approximateing it with a rectangle of height f(x) and change x. As x approaches 0 the rectangle becomes very thin. and it approximates the curve better and better.

The area of that approximating triangle is dx times f(x) . Taking the derivating under and interval is taking summing those rectangle of that size over and over again inside that place.

This thing represent a tiny change in the direction of x on a graph you can show it using the thing we know as tangent

It approximates the area well

We have seen this product before remember? How we have derivative of a function times a changes in x result in the change of that function itself.

The fundamental theorem of calculus tells us that if we take the derivative of a function times change in x then we get the original funtion back.

AI clean up

1. Functions as Mappings

A function (f\colon A\to B) assigns each element of (A) to exactly one element of (B).
Example: if you double every penny you bet, then (f(p)=2p).

2. Visualizing on the Cartesian Plane

– Plot the points ((p,,f(p))).
– If (p=2), then (f(2)=4); if (p=3), then (f(3)=6).
– The average rate of change over ([p,p+h]) is
[ \frac{f(p+h)-f(p)}{(p+h)-p} = \frac{2(p+h)-2p}{h} = 2. ]

3. Rate of Change and Velocity

– A car traveling at constant speed covers 20 km in 1 h.
– Its average rate of change (speed) is
[ \frac{\Delta x}{\Delta t} = \frac{20\ \text{km}}{1\ \text{h}} = 20\ \text{km/h}. ]

4. When Average Fails

If speed varies on ([t_0,t_1]), the average (\Delta x/\Delta t) hides those fluctuations.

5. Instantaneous Rate of Change

– To capture changing speed, we take the limit
[ f’(p) = \lim_{h\to0} \frac{f(p+h)-f(p)}{h}. ]
– For a bullet, the instantaneous velocity at impact is more informative than any average.

6. Leibniz Notation

[ \frac{d}{dx}f(x) = f’(x), \qquad df = f’(x),dx. ]
Here (dx) is an infinitesimal change in (x), and (df) the corresponding change in (f).

7. Connection to Integration

– The product (f’(x),dx) can be seen as the area of a rectangle of height (f’(x)) and width (dx).
– Summing these areas gives the integral: [ \int_{a}^{b} f’(x),dx = f(b)-f(a). ]
– This is the Fundamental Theorem of Calculus.

8. Next Steps

– Sketch a polynomial curve and illustrate Riemann rectangles of height (f(x)) over ([a,b]).
– Show how letting the partition width (\Delta x\to0) yields the exact area.