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AVprozaik [17]
2 years ago
14

A) Use the limit definition of derivatives to find f’(x)

Mathematics
1 answer:
Ann [662]2 years ago
8 0
<h3>1)</h3>

\text{Given that,}\\\\f(x) =  \dfrac{ 1}{3x-2}\\\\\text{First principle of derivatives,}\\\\f'(x) = \lim \limits_{h \to 0} \dfrac{f(x+h) - f(x) }{ h}\\\\\\~~~~~~~~= \lim \limits_{h \to 0} \dfrac{\tfrac{1}{3(x+h) - 2} - \tfrac 1{3x-2}}{h}\\\\\\~~~~~~~~= \lim \limits_{h \to 0}  \dfrac{\tfrac{1}{3x+3h -2} - \tfrac{1}{3x-2}}{h}\\\\\\~~~~~~~~= \lim \limits_{h \to 0} \dfrac{\tfrac{3x-2-3x-3h+2}{(3x+3h-2)(3x-2)}}{h}\\\\\\

       ~~~~~~~= \lim \limits_{h \to 0} \dfrac{\tfrac{-3h}{(3x+3h-2)(3x-2)}}{h}\\\\\\~~~~~~~~= \lim \limits_{h \to 0} \dfrac{-3h}{h(3x+3h-2)(3x-2)}\\\\\\~~~~~~~~=-3 \lim \limits_{h \to 0} \dfrac{1}{(3x+3h-2)(3x-2)}\\\\\\~~~~~~~~=-3 \cdot \dfrac{1}{(3x+0-2)(3x-2)}\\\\\\~~~~~~~~=-\dfrac{3}{(3x-2)(3x-2)}\\\\\\~~~~~~~=-\dfrac{3}{(3x-2)^2}

<h3>2)</h3>

\text{Given that,}~\\\\f(x) = \dfrac{1}{3x-2}\\\\\textbf{Power rule:}\\\\\dfrac{d}{dx}(x^n) = nx^{n-1}\\\\\textbf{Chain rule:}\\\\\dfrac{dy}{dx} = \dfrac{dy}{du} \cdot \dfrac{du}{dx}\\\\\text{Now,}\\\\f'(x) = \dfrac{d}{dx} f(x)\\\\\\~~~~~~~~=\dfrac{d}{dx} \left( \dfrac 1{3x-2} \right)\\\\\\~~~~~~~~=\dfrac{d}{dx} (3x-2)^{-1}\\\\\\~~~~~~~~=-(3x-2)^{-1-1} \cdot \dfrac{d}{dx}(3x-2)\\\\\\~~~~~~~~=-(3x-2)^{-2} \cdot 3\\\\\\~~~~~~~~=-\dfrac{3}{(3x-2)^2}

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Fifteen 7th graders were asked how many hours they spend doing homework each week and how many hours they spend watching televis
timama [110]

Answer:

A.

  • 18 > 14, true

B.

  • 12 - 16 = -4, true

C.

<u>Mean:</u>

  • Homework = (4 + 6*2+10*3+12*2+14*4+16*2+18)/15 = 11.73
  • Television = (0 + 10*3+12*1+16*3+18*4+20*3)/15 = 14.8

<u>The difference:</u>

  • 14.8 - 11.73 > 1, NOT true

D.

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3 years ago
What value of B will make the product equal of 17 in the following expression: 17xb/7
Katyanochek1 [597]

The value B is 7. If we apply B=7 the answer is 17 for the expression 17×\frac{b}{7}

Step-by-step explanation:

            The given expression is 17×\frac{b}{7}  

Step:1

            Form an equation from given data,

            17×\frac{b}{7}= 17..............................(1)

                                                         (∵ product equal to 17)

Step:2

            Check for alternatives by apply the value of B in eqn (1)

            Assume be value of B for check the equation

            B=1

            Equation becomes,

            17×\frac{1}{7} = 2.4286 ≠ 17

            Take, B=2,

            17×\frac{2}{7} = 4.8571 ≠ 17

            Take, B=3,

            17×\frac{3}{7} = 7.2857 ≠ 17

similarly,

            B=7,

            17×\frac{7}{7} = 17                       (∵ The product equals to 17)

Result:

         So, the answer for B is 7.

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Step-by-step explanation:

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in 100 years they will be able to withdraw $131.50

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