Mathematics, 14.08.2020 01:01 Brock2002

Find an equation of the tangent to the curve at the given point by both eliminating the parameter and without eliminating the parameter. x = 5 + ln(t), y = t2 + 2, (5, 3)

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Answer from: 00015746

Step-by-step explanation:

Given that:

x = 5 + In (t)

y = t^2+2

At point (5,3)

To find an equation of the tangent to the curve at the given point,

By without eliminating the parameter

$\dfrac{dx}{dt}= \dfrac{1}{t}$

$\dfrac{dy}{dt}= 2t$

$\dfrac{dy}{dx}= \dfrac{ \dfrac{dy}{dt} }{\dfrac{dx}{dt} }$

$\dfrac{dy}{dx}= \dfrac{ 2t }{\dfrac{1}{t} }$

$\dfrac{dy}{dx}= 2t^2$

$\dfrac{dy}{dx}_{ (5,3)}= 2t^2_{ (5,3)}$

t² + 5 = 4

t² = 4 - 5

t² = - 1

Then;

$\dfrac{dy}{dx}_{ (5,3)}= -2$

The equation of the tangent is:

y -y_1 = m(x-x_1)

(y-3 )= -2(x - 5)

y - 3 = -2x +10

y = -2x + 7

y = 2x - 7

By eliminating the parameter

x = 5 + In(t)

In(t) = 5 - x

$t =e^{x-5}$

$y = (e^{x-5})^2+5$ $y = (e^{2x-10})+5$

$\dfrac{dy}{dx} = 2e^{2x-10}$

$\dfrac{dy}{dx}_{(5,3)} = 2e^{10-10}$

$\dfrac{dy}{dx}_{(5,3)} = 2$

The equation of the tangent is:

y -y_1 = m(x-x_1)

(y-3 )= -2(x - 5)

y - 3 = -2x +10

y = -2x + 7

y = 2x - 7

Answer from: Quest

answer: e 9, 5.75

step-by-step explanation:

step 1

find the variation ab in the x coordinate

delta x=(x2-x1)=10-> 8 units

find the variation ab in the y coordinate

delta y=(y2-y1)=6-> 2 units

step 2

find the coordinates point c

x coordinate

cx=ax+delta x*2/> cx=2+8*(2/> cx=3

y coordinate

cy=ay+delta y*2/> cy=4+2*(2/> cy=4.25

the coordinates of point c are (3,4.25)

step 3

find the coordinates point d

x coordinate

dx=ax+delta x*4/> dx=2+8*(4/> dx=4

y coordinate

dy=ay+delta y*4/> dy=4+2*(4/> dy=4.5

the coordinates of point d are (4, 4.5)

step 4

find the coordinates point e

x coordinate

ex=ax+delta x*4/> ex=2+8*(6/> ex=5

y coordinate

ey=ay+delta y*4/> ey=4+2*(6/> ey=4.75

the coordinates of point e are (5, 4.75)

step 5

find the coordinates point f

x coordinate

fx=ax+delta x*4/> fx=2+8*(8/> fx=6

y coordinate

fy=ay+delta y*4/> fy=4+2*(8/> fy=5

the coordinates of point f are (6,5)

step 6

find the coordinates point g

x coordinate

gx=ax+delta x*4/> gx=2+8*(10/> gx=7

y coordinate

gy=ay+delta y*4/> gy=4+2*(10/> gy=5.25

the coordinates of point g are (7,5.25)

step 7

find the coordinates point h

x coordinate

hx=ax+delta x*4/> hx=2+8*(12/> hx=8

y coordinate

hy=ay+delta y*4/> hy=4+2*(12/> hy=5.5

the coordinates of point h are (8,5.5)

step 8

find the coordinates point i

x coordinate

ix=ax+delta x*4/> ix=2+8*(14/> ix=9

y coordinate

iy=ay+delta y*4/> iy=4+2*(14/> iy=5.75

the coordinates of point i are (9,5.75)

- r3ktforgood ☕

Answer from: Quest

1/3

step-by-step explanation:

to find this, you can divide 12 by 4 which is 3 or you can multiply 4 until you reach 12 (4x3=12). both should give you 3, meaning theres three times less bananas than there are oranges.

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