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NNADVOKAT [17]
3 years ago
11

Which expression was simplified

Mathematics
2 answers:
inessss [21]3 years ago
8 0
First one is the correct answer
Ber [7]3 years ago
7 0
The first one is simplified!
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Let f(x) = 1/x^2 (a) Use the definition of the derivatve to find f'(x). (b) Find the equation of the tangent line at x=2
Verdich [7]

Answer:

(a) f'(x)=-\frac{2}{x^3}

(b) y=-0.25x+0.75

Step-by-step explanation:

The given function is

f(x)=\frac{1}{x^2}                  .... (1)

According to the first principle of the derivative,

f'(x)=lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}

f'(x)=lim_{h\rightarrow 0}\frac{\frac{1}{(x+h)^2}-\frac{1}{x^2}}{h}

f'(x)=lim_{h\rightarrow 0}\frac{\frac{x^2-(x+h)^2}{x^2(x+h)^2}}{h}

f'(x)=lim_{h\rightarrow 0}\frac{x^2-x^2-2xh-h^2}{hx^2(x+h)^2}

f'(x)=lim_{h\rightarrow 0}\frac{-2xh-h^2}{hx^2(x+h)^2}

f'(x)=lim_{h\rightarrow 0}\frac{-h(2x+h)}{hx^2(x+h)^2}

Cancel out common factors.

f'(x)=lim_{h\rightarrow 0}\frac{-(2x+h)}{x^2(x+h)^2}

By applying limit, we get

f'(x)=\frac{-(2x+0)}{x^2(x+0)^2}

f'(x)=\frac{-2x)}{x^4}

f'(x)=\frac{-2)}{x^3}                         .... (2)

Therefore f'(x)=-\frac{2}{x^3}.

(b)

Put x=2, to find the y-coordinate of point of tangency.

f(x)=\frac{1}{2^2}=\frac{1}{4}=0.25

The coordinates of point of tangency are (2,0.25).

The slope of tangent at x=2 is

m=(\frac{dy}{dx})_{x=2}=f'(x)_{x=2}

Substitute x=2 in equation 2.

f'(2)=\frac{-2}{(2)^3}=\frac{-2}{8}=\frac{-1}{4}=-0.25

The slope of the tangent line at x=2 is -0.25.

The slope of tangent is -0.25 and the tangent passes through the point (2,0.25).

Using point slope form the equation of tangent is

y-y_1=m(x-x_1)

y-0.25=-0.25(x-2)

y-0.25=-0.25x+0.5

y=-0.25x+0.5+0.25

y=-0.25x+0.75

Therefore the equation of the tangent line at x=2 is y=-0.25x+0.75.

5 0
3 years ago
Find the value of k so that the graph of 13y + kx = 4 and the line containing the points (5, -8) and (2, 4) are parallel.
guajiro [1.7K]

the graph of 13y + kx = 4 and the line containing the points (5, -8) and (2, 4) are parallel.

We find the slope of parallel line using two given points

(5, -8) and (2, 4)

Slope formula is

slope = \frac{y_2-y_1}{x_2-x_1}

slope = \frac{4-(-8)}{2-5}

slope = \frac{12)}{-3}=-4

so slope = -4

Slope of any two parallel lines are always equal

Lets find the slope of the equation 13y + kx = 4

Subtract kx on both sides

13 y = -kx + 4

Divide both sides by 13

y= \frac{-kx}{13} + \frac{4}{13}

Now slope = -k/13

We know slope of parallel lines are same

So the slope of 13y + kx = 4  is also -4

Hence we equation the slope and find out k

\frac{-k}{13}=-4

Multiply by 13 on both sides and divide by -1

k = 52

the value of k = 52

4 0
3 years ago
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the moon is about 240000 miles from earth what is this distance written as a whole number multiplied by a power of ten
miskamm [114]
24 x 10^4, the 4 comes from with how many jumps it took to land on a whole number from 240,000.  
4 0
3 years ago
From $90 to $39.<br> What is the percent of change? (round to nearest WHOLE percent)
DENIUS [597]
The percentage would be 51% but if you want to round it the answer is 50%
7 0
3 years ago
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18)<br> 46°<br> 47°<br> ?<br> A) 89°<br> C) 159°<br> B) 87°<br> D) 97°
kvv77 [185]

Answer:

89 159 87 23=165237890

5 0
3 years ago
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