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grandymaker [24]
3 years ago
6

Given () = 2, = 1, and ∆ = 0.6:

Mathematics
1 answer:
Scorpion4ik [409]3 years ago
3 0

Answer:

a)

\triangle y = 1.56

b) Dy=1.2

c) See attachment

Step-by-step explanation:

a)Let

\triangle x

be a small change in x, then the corresponding change in y is

\triangle y= f(x  +  \triangle x) - f(x)

We have

f(x) =  {x}^{2}

This implies that:

\triangle y= (x  +  \triangle x)^{2}  -  {x}^{2}

We substitute

x = 1 \: and \:  \:  \triangle x = 0.6

This gives

\triangle y= (1 + 0.6)^{2}  -  {1}^{2}  \\ \triangle y =  {1.6}^{2}  - 1 = 1.56

b) Let

y = f(x)

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

This implies that:

dy = f'(x)dx

with dx=0.6 , and x=1,

dy = 2x dx

dy = 2(1)(0.6) = 1.2

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Answer:

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Step-by-step explanation:

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Also,

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3 years ago
Write the Slope- Intercept equation of the line that passes through the given point and is perpendicular to the given line (0, -
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Answer:

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Step-by-step explanation:

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Answer:

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Step-by-step explanation:

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Find the slope of the tangent line to the given polar curve at the point specified by the value of θ
MariettaO [177]

The given curve has equation

<em>r(θ)</em> = 9 + 8 cos(<em>θ</em>)

and its derivative is

d<em>r</em>/d<em>θ</em> = -8 sin(<em>θ</em>)

When <em>θ</em> = <em>π</em>/3, we have <em>r</em> (<em>π</em>/3) = 13, and d<em>r</em>/d<em>θ</em> (<em>π</em>/3) = -4√3.

Differentiate these with respect to <em>θ</em> :

d<em>y</em>/d<em>θ</em> = d<em>r</em>/d<em>θ</em> sin(<em>θ</em>) + <em>r(θ)</em> cos(<em>θ</em>)

d<em>x</em>/d<em>θ</em> = d<em>r</em>/d<em>θ</em> cos(<em>θ</em>) - <em>r(θ</em>) sin(<em>θ</em>)

In polar coordinates, we have

<em>y(θ)</em> = <em>r(θ)</em> sin(<em>θ</em>)

<em>x(θ)</em> = <em>r(θ)</em> cos(<em>θ</em>)

and when <em>θ</em> = <em>π</em>/3, we have <em>y</em> (<em>π</em>/3) = 13√3/2 and <em>x</em> (<em>π</em>/3) = 13/2.

The slope of the tangent line to the curve is d<em>y</em>/d<em>x</em>. By the chain rule,

d<em>y</em>/d<em>x</em> = d<em>y</em>/d<em>θ</em> • d<em>θ</em>/d<em>x</em> = (d<em>y</em>/d<em>θ</em>) / (d<em>x</em>/d<em>θ</em>)

d<em>y</em>/d<em>x</em> = (d<em>r</em>/d<em>θ</em> sin(<em>θ</em>) + <em>r(θ)</em> cos(<em>θ</em>)) / (d<em>r</em>/d<em>θ</em> cos(<em>θ</em>) - <em>r(θ</em>) sin(<em>θ</em>))

When <em>θ</em> = <em>π</em>/3, the slope is

d<em>y</em>/d<em>x</em> = (-4√3 sin(<em>π</em>/3) + 13 cos(<em>π</em>/3)) / (-4√3 cos(<em>π</em>/3) - 13 sin(<em>π</em>/3))

d<em>y</em>/d<em>x</em> = (-4√3 (√3/2) + 13 (1/2)) / (-4√3 (1/2) - 13 (√3/2))

d<em>y</em>/d<em>x</em> = - 1/(17√3)

So, the tangent line has slope -1/(17√3) and passes through (13/2, 13√3/2). Using the point-slope formula, its equation is

<em>y</em> - 13√3/2 = -1/(17√3) (<em>x</em> - 13/2)

<em>y</em> = -(<em>x</em> - 338)/(17√3)

4 0
2 years ago
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