How do you use the limit definition to find the slope of the tangent line to the graph $f(x) = 4- x^2$ at (2,0)?

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Answer 1 · 2016-11-06T14:12:20

The slope of the tangent line is $-4$ .

When dealing with "limit definition" problems, use the formula $f'(x) = lim_(h->0) (f(x + h) - f(x))/h$ to find the derivative.

$f'(x) = lim_(h->0) (f(x + h) - f(x))/h$

$f'(x) = lim_(h->0) (4 - (x + h)^2 - (4 - x^2))/h$

$f'(x) = lim_(h->0) (4 - x^2 - 2xh - h^2 - 4 + x^2)/h$

$f'(x)= lim_(h-0) (-2xh + h^2)/h$

$f'(x) = lim_(h->0) (h(-2x + h))/h$

$f'(x) = lim_(h-> 0) -2x + h$

$f'(x) = -2x + 0$

$f'(x) = -2x$

We can now substitute the $x$ coordinate of your point into the derivative to find the slope the tangent line.

$f'(x) = -2(2)$

$f'(x) = -4$

Hence, the slope of the tangent line is $-4$ .

Hopefully this helps!

How do you use the limit definition to find the slope of the tangent line to the graph f(x) = 4- x^2f(x) = 4- x^2 at (2,0)?