Mathematics
seteph12
12

For what values of a and b is the line -2x + y = b tangent to the curve y=ax^3 when x=5? No decimal answers.

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(2) Answers
natedog4

Hello, [latex]y=2x+b \\\\ y=ax^3\\\\ slope\ =\ 3ax^2\\ if \ x=5\ then\ slope=75a\ =2\\\\ a= \dfrac{2}{75}\ and \ y= \dfrac{2}{75}* x^3\\\\ if \ x=5\ then\ y=\dfrac{2}{75}* 5^3= \dfrac{10}{3} \\\\ \dfrac{10}{3}=2*5+b\\\\ b= \dfrac{-20}{3} [/latex]

ElizbethLiddle208

the tangent of the curve is determined by getting the first derivative of teh equation of the curve and substituting with  the given data. in this case, the derivative of y=ax^3 is y' = 3a x^2. when x is equal to 5, y' = 3a (25) = 75 a x=5; y = 125 a y-y1 = m(x-x1) y-125 a = 75 a (x-5) y = 75 ax -500 -2x + y = b -2(75/2) x  + y = -500 a = 75/2 b = -500

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