Mathematics, 04.10.2021 01:00 scastillo8

Help me with this one.

Help me with this one.............

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Answer from: mrflexington77

(i)

$\displaystyle \large{f'(x) = 6x - 2}$

(ii/(a))

$\displaystyle \large{y' = 10 {x}+ 3 {x}^{2} - 28{x}^{3} }$

(ii/(b))

$\displaystyle \large{y' = 20 {x}^{ 3} + 90{x}^{2} + 120x +40}$

Step-by-step explanation:

( 1 )

For first question, we have to use first principal or limit definition to find the derivative of 3x^2-2x.

Limit Definition of Derivative

$\displaystyle \large{f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} }$

Our f(x) is 3x^2-2x.

Our f(x+h) is 3(x+h)^2-2(x-h)

$\displaystyle \large{f'(x) = \lim_{h \to 0} \frac{[3 {(x + h)}^{2} - 2(x + h)]- (3 {x}^{2} - 2x) }{h} }$

Expand the expressions and simplify:-

$\displaystyle \large{f'(x) = \lim_{h \to 0} \frac{[3 ( {x}^{2} + 2xh + {h}^{2}) - 2x - 2h]- 3 {x}^{2} + 2x}{h} } \\ \displaystyle \large{f'(x) = \lim_{h \to 0} \frac{3 {x}^{2} + 6xh + 3{h}^{2} - 2x - 2h - 3 {x}^{2} + 2x}{h} } \\ \displaystyle \large{f'(x) = \lim_{h \to 0} \frac{6xh + 3 {h} ^{2} - 2h}{h} }$

Cancel all h's.

$\displaystyle \large{f'(x) = \lim_{h \to 0} \frac{6xh + 3 {h} ^{2} - 2h}{h} } \\ \displaystyle \large{f'(x) = \lim_{h \to 0} 6x + 3h - 2}$

Substitute h = 0

$\displaystyle \large{f'(x) = \lim_{h \to 0} 6x + 3h - 2} \\ \displaystyle \large{f'(x) = \lim_{h \to 0} 6x + 3(0) - 2} \\ \displaystyle \large{f'(x) = \lim_{h \to 0} 6x- 2} \\ \displaystyle \large{f'(x) = 6x - 2}$

Therefore, the derivative of 3x^2-2x is 6x-2 by first principal.

( 2 )

For the second question, there are two sub-questions. Assume that you can use any formulas to differentiate instead of using first principal.

(a)

We are given the function:

$\displaystyle \large{y = 5 {x}^{2} + {x}^{3} - 7 {x}^{4} }$

Our formula for derivative of polynomial function is:-

$\displaystyle \large{y = a {x}^{n} \longrightarrow y' = na {x}^{n - 1} }$

Apply the formula:-

$\displaystyle \large{y = 5 {x}^{2} + {x}^{3} - 7 {x}^{4} } \\ \displaystyle \large{y' = 2(5) {x}^{2 - 1} + 3 {x}^{3 - 1} - 4(7) {x}^{4 - 1} } \\ \displaystyle \large{y' = 10 {x}+ 3 {x}^{2} - 28{x}^{3} }$

If you want it in factored form then:-

$\displaystyle \large{y' = 10 {x}+ 3 {x}^{2} - 28{x}^{3} } \\ \displaystyle \large{y' = x(10 + 3 {x}- 28{x}^{2} )}$

Alternative

You can factor the expression then apply product rules.

Product Rules

$\displaystyle \large{y = f(x)g(x) \longrightarrow y' = f' (x)g(x) + g' (x)f(x)}$

Therefore:-

$\displaystyle \large{y = 5 {x}^{2} + {x}^{3} - 7 {x}^{4} } \\ \displaystyle \large{y = x(5 {x} + {x}^{2} - 7 {x}^{3} )} \\ \displaystyle \large{y' = x'(5x + x^{2} - {7x}^{3} ) + x(5x + {x}^{2} - 7 {x}^{3} )'} \\ \displaystyle \large{y' = 5x + x^{2} - {7x}^{3} + x(5+ 2 {x} - 21 {x}^{2}) } \\ \displaystyle \large{y' = 5x + x^{2} - {7x}^{3} + 5x + 2 {x}^{2} - 21 {x}^{3} } \\ \displaystyle \large{y' = 3 {x}^{2} + 10x - 28 {x}^{3} }$

( b )

For our second sub-question, you can apply product rules instead of expanding in and differentiate.

$\displaystyle \large{y = 5 x {(x + 2)}^{3} } \\ \displaystyle \large{y' = 5 x ' {(x + 2)}^{3} + 5x((x + 2) ^{3})' } \\ \displaystyle \large{y' = 5 {(x + 2)}^{3} + 5x(3(x + 2) ^{2}) }$

Recall cubic formula:

$\displaystyle \large{ {(x + y)}^{3} = {x}^{3} + 3 {x}^{2} y + 3x {y}^{2} + {y}^{3} }$

Therefore:-

$\displaystyle \large{y' = 5 ( {x}^{3} + 6 {x}^{2} + 12x + 8) + 5x(3( {x}^{2} + 4x + 4) } \\ \displaystyle \large{y' = 5 {x}^{3} + 30 {x}^{2} + 60x + 40+ 5x(3{x}^{2} + 12x + 12) } \\ \displaystyle \large{y' = 5 {x}^{3} + 30 {x}^{2} + 60x + 40 + 15 {x}^{3} + {60x}^{2} + 60x } \\ \displaystyle \large{y' = 20 {x}^{ 3} + 90{x}^{2} + 120x +40}$

Answer from: Quest

the most boring ever written

step-by-step explanation:

i am writing something. yes, i plan to make it the most boring thing ever written. i go to the store. a car is parked. many cars are parked or moving. some are blue. some are tan. they have windows. in the store, there are items for sale. these include such things as soap, detergent, magazines, and lettuce. you can enhance your life with these products. soap can be used for bathing, be it in a bathtub or in a shower. apply the soap to your body and rinse. detergent is used to wash clothes. place your dirty clothes into a washing machine and add some detergent as directed on the box. select the appropriate settings on your washing machine and you should be ready to begin. magazines are stapled reading material made with glossy paper, and they cover a wide variety of topics, ranging from news and politics to business and stock market information. some magazines are concerned with more recreational topics, like sports card collecting or different kinds of hairstyles. lettuce is a vegetable.

Answer from: Quest

no association

step-by-step explanation:

if you have eyes and you can use them efficiently you can see that it just looks like random dots like do a line of best fit and like they're so spread out it's such a weak association you might as well consider it nonexistent

Which statement best describes the association between variable x and variable y? perfect negative

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