Based on the graph f(x)f\left(x\right)f(x), describe all points where the derivative f′(x)f^{\prime}\left(x\right)f′(x)would have a jump.

x = − 1 x=-1 x = − 1 and x = 0.5 x=0.5 x = 0.5

Based on the graph f ( x ) f\left(x\right) f ( x ) , describe all points where the derivative f ′ ( x ) f^{\prime}\left(x\right) f ′ ( x ) would have a jump.

So in this problem, we're told based on the graph, f of x, describe all the points where the derivative f prime of x would have a jump. So this would, in essence, mean there's a discontinuity in the derivative graph, and recall that this happens whenever the function is discontinuous at a point or has a sharp turn. Now looking at this function here, I can trace this whole function without picking up the pen. And because of that, that means that this is continuous everywhere, but it's not necessarily going to have a derivative that exists everywhere because we need to watch out for sharp turns or corners as well. Well, I can see that there's a sharp corner that appears right about here, and I can see that there's a sharp corner that appears right there. So what that means is at x equals negative one and at x equals positive 0.5 or 1 half, there is going to be a jump in our derivative graph, and that is going to be the solution to this problem. So I hope you found this helpful, and let's move on.

Based on the graph f ( x ) f\left(x\right) f ( x ) , describe all points where the derivative f ′ ( x ) f^{\prime}\left(x\right) f ′ ( x ) would have a jump.

x = − 1 x=-1 x = − 1 and x = 0.5 x=0.5 x = 0.5

x = − 1.5 x=-1.5 x = − 1.5 , x = − 1 x=-1 x = − 1 , and x = 0.5 x=0.5 x = 0.5

Derivative f ′ ( x ) f^{\prime}\left(x\right) f ′ ( x ) has no jumps

2. Intro to Derivatives

Basic Graphing of the Derivative

Calculus

2. Intro to Derivatives

Basic Graphing of the Derivative

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Multiple Choice

Based on the graph $f\left(x\right)$ , describe all points where the derivative $f^{\prime}\left(x\right)$ would have a jump.

$x=-1.5$ , $x=-1$ , and $x=0.5$

$x=-1$ and $x=0.5$

$x=-1.5$

Derivative $f^{\prime}\left(x\right)$ has no jumps

Verified step by step guidance

To determine where the derivative f'(x) has jumps, we need to look for points on the graph of f(x) where there is a sudden change in the slope. These are typically points where the graph has sharp corners or discontinuities.

Examine the graph at x = -1.5. The graph appears smooth and continuous at this point, indicating that there is no jump in the derivative f'(x) here.

Next, examine the graph at x = -1. At this point, the graph has a sharp corner, indicating a sudden change in the slope. This suggests that the derivative f'(x) has a jump at x = -1.

Now, examine the graph at x = 0.5. Similar to x = -1, there is a sharp corner at this point, indicating another sudden change in the slope. Therefore, the derivative f'(x) has a jump at x = 0.5.

Finally, confirm that the graph is smooth and continuous at x = -1.5, as previously noted, ensuring that the derivative f'(x) does not have a jump at this point.

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