Sketch the graph of a function that is continuous at x=5 but not differentiable at x=5.
horizontal till x=5, then fromx=5 upward sloping line.
Sure, let's see if we can create a "differential dilemma" for you! Imagine a function that is continuous at x=5 but not differentiable at x=5. Let's call this peculiar function "Discombobulated Derivative."
To depict the graph of Discombobulated Derivative, let's start by drawing a smooth curve that passes through x=5. We can make it seem harmless at first, lulling the unsuspecting mathematician into a false sense of security.
But here comes the twist! Right at x=5, we will add a mischievous clown nose to the graph. This nose symbolizes the lack of differentiability. You'll know you've reached the spot where the function becomes a bit "derange-y" in terms of its differentiability.
So, picture a graph with a beautiful curve that stays calm and collected on each side of x=5, but when you hit that specific value, there's a sudden funny-looking clown nose sticking out of the graph. It's like the function is saying, "Sorry, I'm just not smooth in this neighborhood!"
Remember, kids: differentiability is a privilege, not a right! Enjoy your clown-botic function and embrace the joy of mathematical mischief!
To sketch the graph of a function that is continuous at x=5 but not differentiable at x=5, we can create a piecewise function that meets these conditions.
Let's define our function f(x) as follows:
f(x) = 2x for x < 5
f(x) = 10 for x = 5
f(x) = 7x for x > 5
Now, let's plot the graph:
- When x is less than 5, f(x) is represented by the line y = 2x. This line has a positive slope and passes through the origin (0,0).
- When x is exactly 5, f(x) is represented by the point (5,10). This gives us a sudden jump in the graph at x = 5, creating a discontinuity.
- When x is greater than 5, f(x) is represented by the line y = 7x. This line also has a positive slope and passes through the point (5,10).
So, the graph of the function would have a straight line with a positive slope to the left of x=5 (excluding the point x=5), a single point at (5,10), and another straight line with a positive slope to the right of x=5.
Here is a rough sketch of the graph:
```
|
| /
| /
| /
|-/------- (5,10)
|
| \
| \
| \
|
|__________________
5
```
Note that at x=5, there is a sharp corner in the graph, indicating a lack of differentiability at that point. The function is continuous everywhere else.
To sketch the graph of a function that is continuous at x=5 but not differentiable at x=5, we need to consider a function that has a sharp corner or a vertical tangent at x=5.
One such example is the absolute value function, f(x) = |x-5|. Let's analyze it step by step:
1. Start by plotting the points and determining the behavior of the function for x < 5 and x > 5:
- For x < 5: Let's choose a few points to the left of x=5, such as x=4, x=3, x=2, and so on. Evaluate f(x) by substituting these x-values into the function. For example, f(4) = |4-5| = |-1| = 1. Join these points with a smooth curve, maintaining the shape of f(x) = |x-5|.
- For x > 5: Repeat the same process for x-values greater than 5, such as x=6, x=7, x=8, and so on. For example, f(6) = |6-5| = |1| = 1. Connect these points with a smooth curve, ensuring that the shape of f(x) = |x-5| remains.
2. Now, let's focus on the behavior at x=5:
- Since the absolute value function has a sharp corner at x=0, we can modify the original function by shifting it horizontally so that the sharp corner occurs at x=5. To do this, we replace x with x-5 in the original function: f(x) = |x-5|.
- Evaluate f(5) by substituting x=5 into the modified function: f(5) = |5-5| = |0| = 0.
- Plot the point (5, 0) on the graph.
The resulting graph should have a "V" shape at x=5, with the point (5, 0) at the vertex of the "V". The function is continuous at x=5 because the left and right limits approach the same value, but it is not differentiable at x=5 because the graph has a sharp corner or a vertical tangent at that point.