If the tangent line to $y = f (x)$ $y = f(x)$ at $(4, 3)$ $(4,3)$ passes through the point $(0, 2)$ $(0,2)$ , Find $f (4)$ $f(4)$ and $f' (4)$ $f'(4)$ ? An explanation would also be very helpful.

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Answer 1 · 2017-03-04T20:59:30

$f(4) = 3$

$f'(4) = 1/4$

The question gives you $f(4)$ already, because the point $(4,3)$ is given. When $x$ is $4$ , $[y = f(x) = ]f(4)$ is $3$ .

We can find $f'(4)$ by finding the gradient at the point $f(4)$ , which we can do because we know the tangent touches both $(4,3)$ and $(0,2)$ .

The gradient of a line is given by rise over run, or the change in $y$ divided by the change in $x$ , or, mathematically,

$m = (y_2-y_1)/(x_2-x_1)$

We know two points on the graph in the question, so effectively we know the two values we need for $y$ and $x$ each. Say that

$(0,2) -> x_1 = 0, y_1 = 2$

$(4,3) -> x_2 = 4, y_2 = 3$

so

$m = (3-2)/(4-0) = 1/4$

which is the gradient.

If the tangent line to y = f(x)y=f(x)y = f(x) at (4,3)(4,3)(4,3) passes through the point (0,2)(0,2)(0,2), Find f(4)f(4)f(4) and f'(4)f'(4)f'(4)? An explanation would also be very helpful.