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Find the equation of the tangent to y = x^4 – x^2 at the point where x=1.
Does the tangent intersect the curve again?

Sagot :

Answer:

[tex]y=2x-2[/tex]

Step-by-step explanation:

The equation of the tangent line can be found by using the formula [tex]y-y_1=m(x-x_1)[/tex] where [tex]m[/tex] is the slope and [tex](x_1,y_1)[/tex] are the coordinate points of the line.

Therefore, we'll need to find the slope of [tex]y=x^4-x^2[/tex] at the point where [tex]x=1[/tex] by taking its derivative and plugging [tex]x=1[/tex] into the derivative:

[tex]f(x)=x^4-x^2[/tex]

[tex]f'(x)=4x^3-2x[/tex] (Remember to use the Power Rule here!)

[tex]f'(1)=4(1)^3-2(1)[/tex]

[tex]f'(1)=4-2[/tex]

[tex]f'(1)=2[/tex] <-- Our slope here is 2

Now, evaluate [tex]f(1)[/tex] to get [tex]y_1[/tex]:

[tex]f(1)=1^4-1^2[/tex]

[tex]f(1)=0[/tex]

Therefore, the equation of the tangent line is:

[tex]y-0=2(x-1)[/tex]

[tex]y=2x-2[/tex]

See attached graph for a visual reference

View image goddessboi
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