Tangent Time

Calculus Level 4

Let a ( x ) = x 4 + 3 x 2 + 6 a(x) =x^4+3x^2+6 . Let b b be the rate of change of the y y -intercept of the tangent line to a ( x ) a(x) . Find b b when x = 2 x=2 .


The answer is -108.

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1 solution

The slope of the line tangent to y y is 4 x 3 + 6 x 4x^3+6x . Therefore, in slope-intercept form, at some point ( a , f ( a ) ) , y = ( 4 a 3 + 6 a ) x + b → a 4 + 3 a 2 + 6 = 4 a 4 + 6 a 2 + b → b = − 3 a 4 − 3 a 2 + 6 (a,f(a)),y=(4a^{ 3 }+6a)x+b\rightarrow a^4+3a^2+6=4a^4+6a^2+b\rightarrow b=-3a^4-3a^2+6 At a = 2 , b ’ = − 12 a 3 − 6 a = − 108 a=2, b’=-12a^3-6a=-108 .

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