Suppose there is no known formula for

g$($x$),$

but we know that

g$(1)=1$

and

g$($x$)=$

x$2+8$

for all x$.$

$($a$)$

Use a linear approximation to estimate

g$(0\mathrm{.}95)$

and

g$(1\mathrm{.}05).$

g$(0\mathrm{.}95)$g$(1\mathrm{.}05)$

$($b$)$

Are the approximations in part $($a$)$ too large or too small? Explain.

The slopes of the tangent lines are positive but the tangents are becoming less steep, so the tangent lines lie above the curve. Thus, the estimates are too large.The slopes of the tangent lines are positive and the tangents are getting steeper, so the tangent lines lie above the curve. Thus, the estimates are too large.$$The slopes of the tangent lines are positive but the tangents are becoming less steep, so the tangent lines lie below the curve. Thus, the estimates are too small.The slopes of the tangent lines are positive and the tangents are getting steeper, so the tangent lines lie below the curve. Thus, the estimates are too small.