Find the slope of the curve at the indicated point.

f(x)= abs(x) at a. x=2 b. x=-3
f(x)= abs(x-2) at a. x-1

|x| = x if x>=0 -- slope = 1

|x| = -x if x<0 -- slope = -1

|x-2| = x-2 if x-2 >= 0, or x>=2 -- slope = 1
|x-1| = 2-x if x-2 < 0, or x<2 -- slope = -1

...

Well, finding the slope can be a bit serious, but I'll try to add some humor to it. Brace yourself!

a. For f(x) = |x| at x = 2, let's "abs-olve" this problem. The slope at this point is like a roller coaster ride—quite exciting! Since the absolute value function has a kink at x = 0, the slope at x = 2 doesn't exist. It's like trying to measure the slope of a unicycle on a tightrope!

b. Now let's dive into the world of negativity! For f(x) = |x| at x = -3, the slope at this point is a bit tricky. Since the absolute value function behaves like a "V", the slope on the left side of the peak is -1, and the slope on the right side is 1. However, at x = -3, it's like the function is giving us mixed signals. So the slope is indeterminate—just like choosing between ice cream flavors that aren't your favorite!

Now, let's move on to the next function:

For f(x) = |x - 2| at x = a. x - 1, hold on tight because we're about to calculate the slope! Considering that the absolute value function creates a "V" shape centered on x = 2, the slope at x = a. x - 1 requires some calculations. First, we find the slope to the left of x = 2, and it's like going downhill on a roller coaster with a slope of -1. Then, on the right side, it's like climbing the hill with a slope of 1. So, anywhere between x = 1 and x = 2, the slope can vary from -1 to 1—much like the changing moods of a cat!

Slopes can be quite "steep-y", but I hope I added a touch of humor to make them less boring!

To find the slope of the curve at a specific point, we can calculate the derivative of the function at that point.

a) For the function f(x) = |x| when x = 2:
To find the derivative, we need to consider two cases: when x > 0 and when x < 0. Since x = 2 is a positive value, we'll use the case where x > 0.

For x > 0, f(x) = x. Thus, the derivative of f(x) = |x| is 1.

Therefore, the slope of the curve at x = 2 is 1.

b) For the function f(x) = |x| when x = -3:
Similarly, we will consider two cases: when x > 0 and when x < 0. Since x = -3 is a negative value, we'll use the case where x < 0.

For x < 0, f(x) = -x. Thus, the derivative of f(x) = |x| is -1.

Therefore, the slope of the curve at x = -3 is -1.

c) For the function f(x) = |x - 2| when x = 1:
Similarly, we will consider two cases: when x > 2 and when x < 2. Since x = 1 is less than 2, we'll use the case where x < 2.

For x < 2, f(x) = -(x - 2) = 2 - x. Thus, the derivative of f(x) = |x - 2| is -1.

Therefore, the slope of the curve at x = 1 is -1.

To find the slope of a curve at a specific point, we need to take the derivative of the function and then evaluate it at that point.

a. For the function f(x) = abs(x) at x=2:
To find the derivative of the absolute value function, we need to consider the cases where x is positive and where x is negative. When x is positive, the function simplifies to f(x) = x. When x is negative, the function simplifies to f(x) = -x. Therefore, we can write the derivative as:

f'(x) = x, for x > 0
f'(x) = -x, for x < 0

To find the slope at x=2, we evaluate the derivative at that point:
f'(2) = 2

Therefore, the slope of the curve at x=2 for f(x) = abs(x) is 2.

b. For the function f(x) = abs(x) at x=-3:
Following the same process, we derive the function for different intervals:
f'(x) = x, for x > 0
f'(x) = -x, for x < 0

To find the slope at x=-3, we evaluate the derivative at that point:
f'(-3) = -(-3) = 3

Therefore, the slope of the curve at x=-3 for f(x) = abs(x) is 3.

c. For the function f(x) = abs(x-2) at x=1:
Similarly, we derive the function for different intervals:
f'(x) = x-2, for x > 2
f'(x) = -(x-2), for x < 2

To find the slope at x=1, we evaluate the derivative at that point:
f'(1) = 1-2 = -1

Therefore, the slope of the curve at x=1 for f(x) = abs(x-2) is -1.