What is the slope of the curve at t=3 assuming that the equations define x and y implicitly as differentiable functions x=f(t) y=g(t), and x=t^5+t, y+4t^5=4x+t^4?

Hagman7v 2022-09-22 Answered
What is the slope of the curve at t=3 assuming that the equations define x and y implicitly as differentiable functions x = f ( t ) , y = g ( t ) , and x = t 5 + t , y + 4 t 5 = 4 x + t 4 ?
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Answers (1)

Adelaide Barr
Answered 2022-09-23 Author has 9 answers
If:
x = t 5 + t
y + 4 t 5 = 4 ( t 5 + t ) + t 4
y = - 4 t 5 + 4 t 5 + 4 t + t 4 = t 4 + 4 t
Now
d x d t = 5 t 4 + 1
d y d t = 4 t 3 + 4
But
d y d x = d y d t d t d x
So
d y d x = 4 t 3 + 4 5 t 4 + 1
For t=3 the slope will be:
d y d x = 4 ( 27 + 4 ) 5 ( 81 + 1 ) = 124 410 = 62 205 = 3.31
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