In a speed vs. time graph where a curve is depicted, why does the slope of a tangent line drawn to the curve represent the instantaneous speed, and not the average speed?
It represents neither, as-written.
The average velocity (or speed , the magnitude of velocity) is the velocity across an entire interval of time for a change in position. The instantaneous velocity is according to one miniscule moment in time, such as 1 femtosecond in the context of several seconds.
Imagine zooming into a position vs. time graph so much that it looks linear. That's the derivative for a position vs. time graph (velocity in some direction). The slope of the tangent line for a velocity vs. time graph is the instantaneous acceleration, not the velocity. Since this slope can be positive or negative, I said velocity rather than speed.
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The slope of a tangent line drawn to a curve on a speed vs. time graph represents the instantaneous speed because it shows the rate at which the object is moving at a specific moment in time. The tangent line touches the curve at a single point, indicating the speed at that exact instant. In contrast, average speed is calculated by dividing the total distance traveled by the total time taken, and it provides an overall measure of speed over a given interval, rather than at a specific moment.
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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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- Find the derivative using first principles? : #sin sqrt(x)#
- How do you find the derivative of #f(x) = 4/(sqrt(x))# using the limit definition?
- How do you find an equation of the tangent line to the graph of #f(x) = e^(x/2) ln(x) # at its inflection point?

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