Tabansi

2021-08-09

a) Verify that f'(x)=6x-2

b) Find the instantaneous rate of change of f(x) at x=-1

Maciej Morrow

Skilled2021-08-10Added 98 answers

Given:

$f\left(x\right)=3{x}^{2}-2x$

a)$f}^{\prime}\left(x\right)=\underset{h\to 0}{lim}\frac{f(x+h)-f\left(x\right)}{h$

$f(x+h)=3{(x+h)}^{2}-2(x+h)$

$=3({x}^{2}+2xh+{h}^{2})-2x-2h$

$=3{x}^{2}+6xh+3{h}^{2}-2x-2h$

And since$f\left(x\right)=3{x}^{2}-2x$

$f(x+h)-f\left(x\right)=3{x}^{2}+6xh+3{h}^{2}-2x-2h-(3{x}^{2}-2x)$

$=6xh+3{h}^{2}-2h.$

$\frac{f(x+h)-f\left(x\right)}{h}=\frac{6xh+3{h}^{2}-2h}{h}=6x+3h-2$

Now the first equation becomes

${f}^{\prime}\left(x\right)=\underset{h\to 0}{lim}(6x+3h-2)$

=6x-2

b) f'(-1)=6(-1)-2=-8

Result:

a) f'(x)=6x-2

b) -8

a)

And since

Now the first equation becomes

=6x-2

b) f'(-1)=6(-1)-2=-8

Result:

a) f'(x)=6x-2

b) -8

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