Question

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

Answer 1

The linear function that approximates the function is

`y=f(1)+f^(\prime)(1)(x-1)`

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

`f(1)=(1)^3+5(1)^2+4=10`

And,

`\begin{gathered} f^(\prime)(x)=3x^2+10x \\ f^(\prime)(1)=3(1)^2+10(1)=13 \end{gathered}`

Then, the linear function is

`\begin{gathered} y=10+13(x-1) \\ y=10+13x-13 \\ y=13x-3 \end{gathered}`