How do you find instantaneous velocity from a position vs. time graph?

Answer 1
In a graph of position vs. time, the instantaneous velocity at any given point #p(x,t)# on the function #x(t)# is the derivative of the function #x(t)# with respect to time at that point.
The derivative of a function at any given point is simply the instantaneous rate of change of the function at that point. In the case of a graph of position (or distance) vs. time, that means that the derivative at a given point #p_0(t_0, x_0)# is the instantaneous rate of change in position (accounting for "positive" and "negative" direction) with respect to time.
As an example, consider a linear distance function (that is, one which can be represented with a line as opposed to a curve). If this were a function of #x# and #y#, with #y# as the dependent variable, then our function in slope-intercept form would take the form #y=mx+b#, where #m# is the slope and #b# is the value of #y# at #x=0#. In this case, #t# is our independent variable and #x# is our dependent, so our linear function would take the form #x(t) = mt+b#.
From algebra, we know that the slope of a line measures the number of units of change in the dependent variable for every single unit of change in the independent variable. Thus, in the line #x(t) = 2t + 5#, for every one unit by which #t# increases, #x# increases by 2 units. If we were to, for example, assign units of seconds to #t# and feet to #x#, then every second that passed (that is, every increase of one second in #t#), position (or distance) would increase by two feet (that is #x# would increase by two feet)
Since our change in distance per unit of change in time will remain the same no matter our starting point #(x_0,t_0)#, in this case we can be assured that our instantaneous velocity is the same throughout. Specifically, it is equal to #m = 2#. Differentiating the function with respect to #t# yields the same answer. Note that this is only identical to our average velocity throughout the function by design: for a non-linear function (such as #x(t) = t^2#) this would not be the case, and we would need to use differentiation techniques to find the derivatives of such functions.
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Answer 2

To find instantaneous velocity from a position vs. time graph, you need to find the slope of the tangent line at the point of interest on the graph. This tangent line represents the velocity at that specific instant. Mathematically, you can find the slope of the tangent line by calculating the derivative of the position function with respect to time at the given time instant. The derivative gives the velocity at that instant. Alternatively, you can calculate the average velocity over a very small time interval around the point of interest. As the time interval approaches zero, the average velocity approaches the instantaneous velocity.

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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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