the following limit represents the derivative of some function f at some number a lim t->1 [(t^4)+t-2]/t-1. f(x)=? a=?
also for what values of x is the tangent line of the graph of f(x)= (6x^3)-(27x^2)-71x-36. parallel to the line y=x+1.6? enter the x values in order, smallest first, to four places of accuracy:
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`lim_(t->1)(t^4+t-2)/(t-1)` first substitute t=h+1 to get
Noting that `(1+h)^4+(1+h)=h^4+4h^3+6h^2+4h+1+1+h`
`=h^4+4h^3+6h^2+5h+2` and `(1)^4+(1)=2`
so if `f(1+h)-f(1))=h^4+4h^3+6h^2+5h`
So our function is `f(x)=x^4+x` and `a=1`
what values of x is the tangent line of the graph of `f(x)= (6x^3)-(27x^2)-71x-36` . parallel to the line `y=x+1.6` ?
`f'(x)=18x^2-54x-71` Take the derivative.
So when is f'(x)=1 (1 is the slope of the line `y=x+1.6` )
`18x^2-54x-71=1` Find when the derivative is =1
`18x^2-54x-72=0` Write in standard form
Divide everything by 18
` x^2-3x-4=0 `
So `x_1 = -1` , `x_2 = 4` By the zero product property
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