$\text{For $f(x)=ax^n$}\\\\\lim\limits_{h\to 0}\dfrac{f(x+h)-f(x)}{h}=\lim\limits_{h\to 0}\dfrac{a(x+h)^n-ax^n}{h}\\\\=\lim\limits_{h\to 0}\dfrac{ax^n+anx^{n-1}h+O(h^2)-ax^n}{h}=\boxed{anx^{n-1}}$

where O(h²) is the series of terms involving h² and higher powers. When divided by h, each term has h as a multiplier, so the series sums to zero when h approaches zero. Of course, if n < 2, there are no O(h²) terms in the expansion, so that can be ignored.

This can be referred to as the power rule.

Note that for the quadratic f(x) = ax^2 +bx +c, the limit of the sum is the sum of the limits, so this applies to the terms individually:

lim[h→0](f(x+h)-f(x))/h = 2ax +b

4. The gradient of 3x^2 is 3(2)x^(2-1) = 6x.

5. The gradient of x^2 +3x +1 is 2x +3.

If you need to "show work" for these problems individually, use the appropriate values for 'a' and 'n' in the above derivation of the power rule.

3 0

3 years ago

Discuss how surface area and volume play a role in marketing products, such as advantages and disadvantages. Give specific examp

lubasha [3.4K]

Big packages drives you to the illusion that the contents are bigger or better which is actually not possible.Even garbage can fill up a huge truck.
Selling something with more volume means you get more for the money. 
Advantage of marketing something as bigger helps you to get them to buy more than the usual.Its a technique to sell more items. 
Disadvantage is they don't need to come back as often to buy again.

5 0

3 years ago