Can you determine the equation of the tangent line of a point on a function by solely using the function's graph?

Answer 1

I would say, that we cannot "determine", but we can estimate.

The graph of a function can never be 100% accurate in real life, unlike a function provided by an expression or expressions.

Estimating is the best we can ever do.

We can only estimate the tangent lines if we are merely estimating the function.

Here is an illustration of a function's graph:

graph{.9995x^2 [-5, 5, 10, -10]}

And supppose we are asked for the equation of the tangent line at the point where #x = 2#.
It is unlikely that anyone will correctly see that the point on the graph has coordinates #(2.3.998)# and the tangent line has equation #y=3.998x-3.998#
We could instead concoct an example in which there is a tiny cusp at #(2,4)#. In this case there would be no tangent line at #(2,4)#
#f(x) = {(x^2,"if",x <= 2),(0.995x^2+0.002,"if",x > 0):}#
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Answer 2

Yes, it is possible to determine the equation of the tangent line of a point on a function by solely using the function's graph. To find the equation of the tangent line, you need to determine the slope of the tangent line at the given point. This can be done by finding the slope of a secant line passing through the given point and a nearby point on the graph. As the nearby point gets closer to the given point, the secant line becomes closer to the tangent line. Once you have the slope of the tangent line, you can use the point-slope form of a linear equation to find the equation of the tangent line.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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