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#?

Answer 1
If: #x=t^5+t# #y+4t^5=4(t^5+t)+t^4# #y=-4t^5+4t^5+4t+t^4=t^4+4t#
Now: #dx/dt=5t^4+1# #dy/dt=4t^3+4#
But: #dy/dx=dy/dtdt/dx#
So: #dy/dx=(4t^3+4)/(5t^4+1)#
For #t=3# the slope will be:
#dy/dx=(4*(27+4))/(5*(81+1))=124/410=62/205=3.31#
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

The user is interested in asking questions about mathematics.To find the slope of the curve at ( t = 3 ), we first need to express ( x ) and ( y ) explicitly in terms of ( t ). Differentiating the second equation with respect to ( t ) and then solving for ( y' ) gives ( y' = \frac{4 - 20t^4}{1} ). Evaluating this at ( t = 3 ) gives ( y' = -236 ). Similarly, differentiating the first equation and solving for ( x' ) gives ( x' = 5t^4 + 1 ), which at ( t = 3 ) is ( x' = 454 ). The slope of the curve at ( t = 3 ) is then ( \frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{-236}{454} = -\frac{118}{227} ).

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7