How do you find the equations of the tangents to that curve #x=3t^2 + 1# and #y=2t^3 + 1# that pass through point (4,3)?

Answer 1

Tangent Line
#y=x-1#

We find the equation first consisting only of x and y by eliminating variable t.

Given #x=3t^2+1" "#first equation and #y=2t^3+1" "#second equation

Use the first equation then substitute its equivalent in the second equation

#x=3t^2+1" "#first equation #t=((x-1)/3)^(1/2)" "#first equation
#y=2t^3+1" "#second equation #y=2(((x-1)/3)^(1/2))^3+1" "#second equation
We now have y in terms of x #y=2((x-1)/3)^(3/2)+1" "#
Solve for the slope #m=dy/dx#
#m=dy/dx=2*3/2*((x-1)/3)^(3/2-1)*d/dx((x-1)/3)+d/dx(1)#
#m=dy/dx=cancel2*cancel3/cancel2*((x-1)/3)^(1/2)*(1/cancel3)+0#
#m=((x-1)/3)^(1/2)" "#at #(4, 3)#
#m=((4-1)/3)^(1/2)" "#
#m=1#
The Tangent Line using #(x_1, y_1)=(4, 3)# and #m=1#
#y-y_1=m(x-x_1)#
#y-3=1(x-4)#
#y=x-4+3#
#y=x-1#
Kindly see the graphs of the curve #y=2((x-1)/3)^(3/2)+1" "# and the tangent line #y=x-1#. graph{(y-2((x-1)/3)^(3/2)-1)(y-x+1)=0[-10, 10, -5,5]}

God bless...I hope the explanation is useful.

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

To find the equations of the tangents to the curve (x = 3t^2 + 1) and (y = 2t^3 + 1) that pass through the point ((4,3)), we follow these steps:

  1. First, we differentiate both equations with respect to (t) to find expressions for (\frac{dy}{dx}).

  2. Then, we find the value of (t) that corresponds to the point of tangency by substituting (x = 4) and (y = 3) into the parametric equations.

  3. Next, we find the slope of the tangent line at that point by substituting the (t) value into the expression for (\frac{dy}{dx}).

  4. After obtaining the slope, we use the point-slope form of the equation of a line to find the equations of the tangents.

By following these steps, we can determine the equations of the tangents to the given curve that pass through the point ((4,3)).

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