Linearization

Approximation of the y value of a function at x=c. By using the tangent line create at a point (a) near x=c, and pug it in.

1. Find the equation of the tangent line of point a while naming it L(x)
2. pug in c and get the answer A
3. develop the notation of $f(c)\approx L(c)=A$
4. If ask if L(c) is a over or under approximation.
   1. If f"(a)<0,f(x)is concave down at x=c and L(a)is an OVER-approximation.
   2. If f"(a)>0,f(x)is concave up at x=c and L(a)is an UNDER-approximation.

(Since f’’(x)>(<) 0 @x=a means the trend of increasing(decreasing) of the derivative of f(x))

Differential

Recall Leibniz’s notation for the derivative function.

$$ \frac {dy} {dx}=f'(x) $$

dy is the differential of y and dx is the differential of x.

Similar to $\Delta y$ and $\Delta x$, which are finite, measurable quantities, the differentials denote a change in respective values, however, they are infinitely small, immeasurable differences approaching zero.

By treating them as individual quantities…

$$ \frac {dy} {dx}=f'(x)=>{dy} =f'(x){dx} $$

The Leibniz’s notation change into the “differential form” of the derivative of y.

Delta and Differential

Related Rates UF (similar???)

In definition, the only difference (if there is any) is their size. Unlike delta, differential are approaching zero; as the difference for two points that is in theory one. So, if $\Delta x$ the distance of the x values is small enough (y values are getting close too, so no big difference), dy will be around $\Delta y$.

$$ \Delta x→dx⇒\Delta y → dy $$

So…$dy\approx \Delta y$