Answered

find the linear function that approximates f(x)= x^3+5x^2+4 at a =1

Answer :

The linear function that approximates the function is

[tex]y=f(1)+f^{\prime}(1)(x-1)[/tex]

So we need to find f(1) and f'(1)

[tex]f(1)=(1)^3+5(1)^2+4=10[/tex]

And,

[tex]\begin{gathered} f^{\prime}(x)=3x^2+10x \\ f^{\prime}(1)=3(1)^2+10(1)=13 \end{gathered}[/tex]

Then, the linear function is

[tex]\begin{gathered} y=10+13(x-1) \\ y=10+13x-13 \\ y=13x-3 \end{gathered}[/tex]

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