The figure shows how a function $f(x)$ and its linear approximation $($i$.$e$.,$ its tangent line$)$ change value when $x$ changes from $x_0$ to $x_0+dx.$

Suppose $f(x)=x^2+6x,x_0=3$ and $dx=0\mathrm{.}01.$ Your answers below need to be very precise, so use many decimal places.

$($a$)$ Find the change $f=f(x_0+dx)-f(x_0).$

$f=$

$($b$)$ Find the estimate $($i$.$e$.,$ the differential$)df=f^{\text{'}}(x_0)dx.$

$df=$

$($c$)$ Find the approximation error $|f-df|.$

Error $=$