Mean and Instant Velocity

class: center, middle, inverse, title-slide

.title[
# Mean and Instant Velocity
]
.subtitle[
## Session 01
]
.author[
### Alejandro Ucan
]
.date[
### 2022-10-15
]

---

# Goals:

* Understand the differences between mean and instant velocity. <br/><br/>
 * The derivative as a instant velocity indicator. <br/><br/>
 * The vectorial version of displacement and velocity.
---
# Problematic.

> A body falls from a building of 500 m height It is evident, body's height `$h$` diminish by the time pass.

> Note that even if `$h$` diminish, the _rate of change_ it is not constant. This is true because the falling _velocity_ is not constant. Why?

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# The mean velocity.

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# Computing the mean velocity.

> These are some known values for the height at the time `$t$` in seconds.

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# Instant velocity.

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# Mathematical formulation for Instant velocity.

> Mean velocity is `$$\overline{v}=\frac{\Delta h}{\Delta t}=\frac{\mbox{displacement}}{\mbox{time}}$$`

If we reduce the time passed to cero (in the limit), we obtain the instant velocity.

`$$v=\lim_{\Delta t\to 0}\overline{v}=\lim_{\Delta t\to 0}\frac{\Delta h}{\Delta t}=\frac{d h}{dt}.$$`
---
# How to compute a derivative as a limit.

> A car is moving on a straight line, if we know that its displacement is given by the function `$f(x)=3x+2.$` Compute the instant velocity.

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# Excercise

> A bird is flying following a straight direction. If we know that the height of the bird is given by the function `$f(x)=1-3x^2$` compute the instant __rate of change__ of the height.

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# Recall

Recall that there are some formulas that ease the derivative computations.

> __Theorem:__ The derivative of the function `$f(x)=x^n$` is given by `$$\frac{d f}{dx}=nx^{n-1}.$$` And also, for a general functions `$f$` and `$g,$` and real numbers `$c,$` we have that `$$\frac{d(f+g)}{dx}=\frac{df}{dx}+\frac{dg}{dx}$$` `$$\frac{d(cf)}{dx}=c\frac{df}{dx}.$$`

__Example:__ If `$f(x)=x^4$` then `$f'(x)=4x^{4-1}=4x^3.$`

---
# Example:

> Compute the derivative of the function `$f(x)=3x^2+2x-1.$`

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# Excercise:

> Find the derivative of the following function `$f(x)=10x^5-5x^4+1.$`

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# The vectorial version.

> The __vectorial curve__ that describe the displacement of the body is written as: `$$\vec{r}(t)=t\mathbf{i}+(500-4.9t^2)\mathbf{j}.$$` <br/>
In this curve, we think that the `$x-$`axis correspond to tiem and the `$y-$`axis to the height. <br/><br/> Note that `$\mathbf{i}$` and `$\mathbf{j}$` are the canonical vectors.

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# And the derivative?

> Well, we can translate the velocity (derivative) of the body using vectors. The derivative will be the derivative of each component of the displacement vector respect to time. `$$\vec{v}(t)=\frac{d t}{dt}\mathbf{i}+\frac{d(500-4.9t^2)}{dt}\mathbf{j}=\mathbf{i}+-9.8t\mathbf{j}.$$`

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# A geometric view of the vectorial velocity

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# Example:

> Find the expression for the vectorial velocity of a car whose displacement (in vectors) is given by the function `$$f(t)=t\mathbf{i}+(3t-1)\mathbf{j}.$$`

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# Exercise:

> Find the expression for the vectorial velocity of a bird whose displacement (in vectors) is given by the function `$$f(t)=t\mathbf{i}+(4t^3-3t^2)\mathbf{j}.$$`