Question
If f(x) = x3 + 7x2 + 8x - 9, find f(4).
✓
Answer
f(x) = x3 + 7x2 + 8x - 9 is a polynomial function. So, it is differentiable every.
$\text{f}'(4)=\lim_\limits{\text{h}\rightarrow0}\frac{\text{f}(4+\text{h})-\text{h}(4)}{\text{h}}$
$=\lim_\limits{\text{h}\rightarrow0}\frac{[(4+\text{h})^3+7(4+\text{h})^2+8(4+\text{h})-9]-[64+112+32-9]}{\text{h}}$
$=\lim_\limits{\text{h}\rightarrow0}\frac{[64+\text{h}^3+48\text{h}+12\text{h}^2+112+7\text{h}^2+56\text{h}+32+8\text{h}-9]-[210-9]}{\text{h}}$
$=\lim_\limits{\text{h}\rightarrow0}\frac{\text{h}^3+19\text{h}^2+112\text{h}+210-9-210+9}{\text{h}}$
$=\lim_\limits{\text{h}\rightarrow0}\frac{\text{h}^3+19\text{h}^2+112\text{h}}{\text{h}}$
$=\lim_\limits{\text{h}\rightarrow0}\frac{\text{h}(\text{h}^2+19\text{h}+112)}{\text{h}}$
$\text{f}'(4)=112$
$\text{f}'(4)=\lim_\limits{\text{h}\rightarrow0}\frac{\text{f}(4+\text{h})-\text{h}(4)}{\text{h}}$
$=\lim_\limits{\text{h}\rightarrow0}\frac{[(4+\text{h})^3+7(4+\text{h})^2+8(4+\text{h})-9]-[64+112+32-9]}{\text{h}}$
$=\lim_\limits{\text{h}\rightarrow0}\frac{[64+\text{h}^3+48\text{h}+12\text{h}^2+112+7\text{h}^2+56\text{h}+32+8\text{h}-9]-[210-9]}{\text{h}}$
$=\lim_\limits{\text{h}\rightarrow0}\frac{\text{h}^3+19\text{h}^2+112\text{h}+210-9-210+9}{\text{h}}$
$=\lim_\limits{\text{h}\rightarrow0}\frac{\text{h}^3+19\text{h}^2+112\text{h}}{\text{h}}$
$=\lim_\limits{\text{h}\rightarrow0}\frac{\text{h}(\text{h}^2+19\text{h}+112)}{\text{h}}$
$\text{f}'(4)=112$
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