mathematics for economists 1

Question

Find the tangent line to the curve at the given point: y+3x²y⁵= 4x+3 at (0,3)

Answer 1

If we let F [x , y ] = y + 3 x² y⁵ − 4x −3

Then d y / d x = - [ F _x ] / [ F _y ]

Hence d y / d x = - [ 6 x y⁵ −4 ] / [1 + 15 x ² y⁴]

Therefore ( d y / d x )_(0,3) = 4

Then the equation of the tangent line y = 4 x +3

Answer 2

1. Use implicit differentiation to find dy/dx.
2. Next, substitute the given point to find the slope of the tangent line.
3. Then find the equation of the tangent line by plugging the slope and the given point and write it in the slope-intercept form.

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Step 1: Use implicit differentiation to find dy/dx.

Given curve equation; y+3x²y⁵= 4x+3

Now differentiate implicitly i.e, (dy/dx)+ 6xy^5+15x^2y^4(dy/dx)=4+0 (Note: Product rule must be applied to the second term in the equation)

Next, make (dy/dx) the subject of the formula shown below:

(dy/dx)[1+15x^2y^4]+6xy^5=4

(dy/dx)[1+15x^2y^4]=4-6xy^5 (Divide both sides by [1+15x^2y^4])

(dy/dx)= (4-6xy^5)/[1+15x^2y^4]

Step 2: Next, substitute the given point to find the slope of the tangent line.

Substitute (0,3) to (dy/dx) equation.

(dy/dx)= (4-6(0)(3)^5)/[1+15(0)^2(3)^4]

= 4

so m=4 aka slope.

Step 3: Then find the equation of the tangent line by plugging the slope and the given point and write it in the slope-intercept form.

First find b. Substitute (0,3) and m=4 to y=mx+b

3=4(0)+b. So b=3

Therefore the equation of the tangent line is y=4x+3