Let f(x)=5sqrt{x}+x^2. Use the linear approximation of f at x=8 to estimate f(7.99).

Answer 1

f(x) = 5√x + x² Let x = 8 and Δx = -0.01 (since 7.99 is 0.01 less than 8)

From the definition of derivative, f'(x) ≈ [f(x+Δx) - f(x)] / Δx.

So, f(x+Δx) ≈ f(x) + f'(x)Δx

Therefore, f(7.99) ≈ f(8) + f'(8)(-0.01) = (5√8 + 64) + (5 / (2√8)+2(8))(-0.01) = 78.142 - 0.169 = 77.97

Answer 2

f(x)=5sqrt{x}+x^2

=5*x^.5+x^2

=2.5x^(-.5)+2x

In this case; multiply/subtract.

When using a fraction as a power; say 2/3; means cubed rt, squared; or (^(1/3))^2.